All of SAT Geometry and Trigonometry

By the end of this video, you'll have learned how to solve every single geometry and trigonometry question on the SAT, which is about 14.2% of the math section according to my latest statistics. Now, for geometry and trigonometry questions, I break it down into six different types of questions, which we're going to go over in order in this video. And each of these types of questions builds off of the knowledge From the previous one. So, to know air and volume, you need to know a bit of similar shapes and right triangles and angles, lines, and triangles

questions. And in case you were curious, I think a lot of people would be interested in this. I also have the distributions of the types of questions between module one and a hard second module. And if you don't know, the SAT is adaptive and there's module one and module two for the reading writing and the math Section. If you do well in module one, they give you a harder second module. If you don't do well in module one, they give you an easier second module. This is just for the hard second module because if you

watch my stuff, you're going to get the hard second module anyways. Okay. First of all, you'll notice the geometry and trigonometry questions in module one surprisingly is higher than it is for module two. You'll see less of these in module two. But the Difference is these are going to be a lot harder questions. You notice the easier types of patterns. There's a lot more of those in module one. There's a lot less of them in module two. But enough of that. Let's jump straight into the first type of question which is angles, lines, and triangles.

For angles, lines, triangle questions, you need to know these seven patterns and be able to recognize them really well. But don't worry because these are pretty Easy and you probably know most of them or all of them already. Okay, the first thing is vertical angles which is basically anytime straight lines intersect each other during that intersection the angles right across from each other are the same. So for example, this angle is the same as this angle because it's right across from each other. This angle is going to be the same as this angle because they're

right Across from each other. Okay? And it doesn't just have to be two lines either. It could be multiple lines. Let's say there's three lines at the intersection. Sorry, it's a bad line. But at the intersection of these lines, like this angle is the same as this angle because it's the angle right across each other. However, this angle is not the same as this angle cuz this angle is not across from that angle. Okay? This angle is across from this Angle. So, it's only these two angles that would be the same. Then, you don't know

for a certain that this is the same as that one. Even if it looks the same, it's not it's not one of the patterns. So, you can't use it. Okay? It just has to be um one of the four patterns that we go over. Next pattern is supplementary angles, which is the idea that if there's a bunch of angles on a line, then all these angles add up to 180°. Okay, so let's say this is only Two. There could be multiple. Okay, let's say there's three angles, right? This angle, this angle, and this angle. If

you add up all three of these angles, they should add up to 180°. Because if you think about it, the angle of a straight line is 180 degrees, right? This from here, that's 180°. So it makes sense that all these should add to 180°. But um just memorize that. Okay, just you should be able to recognize that pretty easily too. Okay, if there's a Bunch of angles on a line, they add up to 180 degrees. And you can use that to find let's say we we know this angle and we know this angle, but we

don't know this angle. Then we can just do 180 minus these two to get this one. Okay, so it's a bit messy, but we'll move on. The next one is corresponding angles. And these only work with parallel lines. Okay? And you know the parallel because the question tells you it. Or also if you see these Symbols which is like arrow marks on the two angles that means these two are parallel to each other. If you see arrow marks and the arrow marks on two lines they're parallel to each other. Okay? Just because they look parallel

doesn't mean they're parallel. The question actually has to say that they're parallel. This is called corresponding angles. Okay. And it's the easy way to memorize this. It's just the corresponding angles where the angles on The same spot on the two different parallel lines are the same. So this angle is the exact same spot, right? The top right of this parallel line and the top right of this parallel line. Okay, these are the exact same spots on two different parallel lines. That means they're the same. It has to be on the same line that intersects both

of them though. If there's a line that intersects both of the parallel lines on the same spot of the intersection Between the two parallel lines, those are the same. So there's another line here. Then you can't say for certain that this angle is the same as like this angle, for example, or this angle is the same as this angle. Unless we were told that these are parallel lines and then we be oh it's it's because this line intersects both of them then oh yeah they are the same spot so it's the same but if we

don't know that these are parallel lines then it's not okay I I'll Make this simple and it doesn't just have to be this specific spot it could be any of them like this angle is the same as this angle this angle is the same as as this angle because they're the same spot right the bottom left corner of this intersection between the two pair of lines these are corresponding angles okay so they're the same now the last pattern I just want you to know this. You technically don't need to memorize this as a separate Pattern

because you can derive this this pattern from the other stuff we went over. And the pattern is alternate interior angles. Interior angles, by the way, is just the angles on the inside. Go figure. Right? So these are the angles on the outside. It doesn't matter. So the angles on the inside, okay, these the inside stuff, okay, the inside angles, the ones that alternate are the same. That's why it's called alternate interior angles. So this angle Is on the inside and this angle is on the inside and they they're alternate like they're right across from each

other, right? They're alternate angles. So they're the same. Okay, you could technically get this from corresponding angles and vertical angles. For example, I know this angle is the same as this angle, right? And I know these two angles are vertical angles because there's two straight lines intersecting each other is obviously the angles Across each other the same. So we already know that these angle is the same as this one. But I just want you to memorize this as a separate pattern just so you can recognize it faster because this is pretty common. You see this

a lot on the SAT. So these are the same angles alternates. Also same thing like this angle is the same as this angle, right? Because they're interior angles and they're alternate. Also, you don't see this a lot, but it technically also Goes for these angles, right? The alternate exterior angles, I guess, are the same as well because it's the same, right? Um this angle is the same as this angle because they're corresponding angles and vertical angles to that one is this. So basically the idea is you can't just guess if angles are the same or

not. You have to prove the angles are the same by using one of these four patterns. If it's not one of this just has to do with lines. Okay? No shapes Involved. Just lines by themselves. If you see lines then you have to know you have to prove it that the those things are the same. You can't just assume they're the same because they look the same on the SAT. Okay? And these two have to do parallel lines. Okay? It only happens if they're parallel lines. These don't really matter. It's just like any anywhere you

see them, right? But this is only for parallel lines. They have to they have To question has to give away that those two lines are parallel to each other. Now, let's go over triangles. And this is just for regularish triangles. This isn't right triangles or anything like that. It's just normal triangle triangles in general. Okay? And you probably already know this, but all the angles in a triangle add up to 180°. This is also in the reference sheet by the way on the the SAT testing software. If you click on reference, the the Reference sheet

pops up and you'll notice at the very bottom it says all the angles in the time pass add to 180. So we know this angle plus this angle plus this angle, right? These three angles and they add 180°. So we just need know two of them to get the dot one. Okay, this is a helpful pattern to know. If you're stuck, you should probably use triangles. If you can find a triangle in there somewhere, which I show example questions on later on, but You just need to know, okay, let's say I know this angle, I

know this angle, I can just do 180 minus these two angles to get the last one because they have to 180°. Okay, that's just like strategy. You should recognize that pattern. Okay, or you should know to look for that. Another thing is isoclesles triangles, which is really common. Isoclesles triangles is just anytime we know two angles or or lengths of the triangle are the same. So, for example, this one, Right? this triangle. So, let me get rid of previous. Okay, this triangle is telling us, by the way, the slashes just means the sides are the

same. Okay, like the number of slashes, right? So, if I had this, this is the same as this one, right? If I had like another triangle and I said one slash there. So, this one has the same amount of slashes as this one. So, that means these two are the same. Same thing if I said like two Slashes here and I said this one has two slashes as well. Since these two have the same amount of slashes and I know this is the same length as this one. Okay, so that's just what the slashes mean

in case you see that. Same thing with like the you might notice me doing the angles over here is the same thing like this has only one arc then this is the same and I put another arc here. So these both have only one arc so they're the same angle. If I said these have two Arcs and this one has two arcs then I know these two angles are the same because they have the same amount of arches. Okay, if you see that on the SAT, then the same. And now we'll do example questions on

that. It it'll be pretty easy to recognize. Okay, so basically we know that these two lines, I guess these two lengths are the same, that means it's triangles. Anytime you see that there's two equal angles or two equal Lengths, then by default, it's an isosesles triangles, which means that two angles are going to be the same. So obviously if you just look at this triangle you should be able to tell okay if if I know these two lengths are the same which two angles of this triangle should be equal to each other and you should

be able to tell are these two right these two angles are the same and that means this one is probably different from these two but these two Are the same but for isocles triangles you just need to know one angle and you have to be really good at finding it so all I need is this one measure of any of the angles here and the fact that it's a isles triangle if I just know one angle on here then I can find the other two let's do practice okay let's Say this was 20°. Okay, we

knew this was 20° and that's it. Find the other angles of this. How would you do it? Cuz so obviously we already know that this Angle is the same as that one. So we know this is also 20°. And we know that all the angles in the triangle add up to 180°. So we can just do 180 minus the two angles that we do know. So I guess - 20 - 20. So - 40, right? So that means that the last angle here adds up to 140 because 180 - 40 is 140, right? So, we

know this angle is 140°. You see, we just need a one and same thing. Let's say, let's erase this and get new leng I guess new angles. Let's say this one was 100°. Okay, what is the angles of the other two? This one's a bit harder, but it's also pretty pretty simple. We know these two angles are the same, right? Because this is the triangles. And we know that all the angles has to add up to 180° is 180 minus 100. So we know these two angles, the angles that are left has to add up

to 80° because the entire thing has to add to 180 and we subtract 100 from there no One in them. So these two left has to add up to 80°. Okay? And we know these angles are the same. So you just do 80 divided by two which is 40. So we know these are four. So basically any time is triangles and you know one of the angles you're able to find the other two. So this is a strategy. So if you see a sauce triangle in the question, you should be thinking to yourself, oh, I

just need to find one of these angles that allows me to get the other two. Okay. And the last thing which is not as common is equilateral triangles, which I I will go over, but this is extremely simple. If you see a equilateral triangle which is just all the angles and lengths are equal and again you don't even need to know it doesn't need to say itself that all the angles the same all the angles are equal you just need to see like all the the lengths are equal that's enough or if you see all

the angles are the same that that's That's enough as well. Basically you just need one of those two. Okay. And like all the line if all the lengths are the same then by default all the angles are the same. That's the idea behind it. But basically equilateral triangle, you should already learn this in in high school hopefully. But all the angles of a equilateral triangle are 60°. So if you see a equilateral triangle, then you don't need to know any angles. You don't need to know a single one of Them because you already know by

default. Like you the question doesn't need to tell you what any of the angles are. But as long as you know it's equilateral triangle, you know by default that all the angles are or 60°. And just to help you memorize that obviously all the angles in triangle has to add up to 180 degrees. And we know all three of these are equal. So this is 180 divided by three is 60. That's how we know all the angles are. So 180 and There's three angles and we know all these angles are the same. 180 divided by

three that equals 60. So that's how we know that all the angles are 60 for equal triangles. It's basic strategy. See that wasn't that hard, right? Just to summarize really quickly before we do practice questions because I'm going to find some pretty hard practice questions. It's just you need to know these seven patterns. Okay, these are patterns. I think of them as patterns. So if you see like a vertical angle with two angles the same, you still want to be think okay I can probably use this and I'll go over strategy when we do practice

question. But you really I think your focus should be around the triangles. Okay, I see a triangle. Oh, I just need to find two of the angles to get the other one. You just look for triangles and then to get the angles that you need to to use these rules, you have to use these. Okay, I'll do example Questions just to show you what I mean. So, let's look at this question. Hopefully, this is big enough. Okay. Yeah. So, I would pause this video and try to solve it yourself. At least figure out how you're

going to get the answer and then unpause it to see how I do it. This is just good practice to get yourself feedback loops for checking to see if the way you you you thought you could solve the question was right or not. Okay. In general strategy wise obviously I would try to understand the question first. But after I understand the question I'm going to look at what I need to find. Okay. All the angles I need to find like what what's the goal and then try to get triangles around here. You notice there's triangles

here. And let me start annotating. So if we see here there's there's a big triangle here. Okay, there's a big triangle there and there's a small triangle there. So There's actually two triangles in here. We can take advantage of that. Okay, so we Oh, we know I I don't think it says let me read the question. I don't think it says that these are triangles or equilateral. So we know that we need to find at least two of the angles here to get the third one. Okay, let me let me read the question cuz I

don't remember the answers. I always remember how to solve it. I just looked hard so I picked it. Okay, par line. So it tells us the Par lines. So immediately you should be thinking, oh corresponding alternate interior angles, right? Basically just corresponding angles. Okay, you can use that. Q and T are intersected by lines R and S. Okay, got it. If A= 43, wait, let me let me get my annotation thing. Okay, so we know these are parallel. Let me just mark that just so you can follow along with this. So we know these are

parallel and O and S. Yeah, these these lines intersect the Parallel lines. A is 43. Okay, immediately I'm thinking like this this big triangle and this is vertical angles that one. So we already have one of the angles of the triangle basically. Okay, and I'll explain that later. But that's 43 B is 122 and we need to find W. Okay, so we need to find this. All right, we just need to find one of these. But since these are the same, I think we just need to find this one and then we can divide it

by Two to get like what W is because we know these are the same, right? So we need to find this one. How do we get this one? Okay, so again, the strategy is you look at what you need to get and try to make triangles. So here's a big triangle I'm thinking because we can probably use this triangle to get this one, right? Because it's supplementary angles, right? 180 180. Um, yeah. So if if you add up these, they have to add up to 180 Supplementary angles. So it's most specifically 180 minus this angle

has to equal these two. So you should immediately be thinking, oh, I can I can use that, right? So let me let me finish this triangle here. That's a really bad triangle, but you get the idea. Okay, so we need we just need to get this angle and this angle and then we can get this angle because that's how a triangle works. We just need two of the angles. We already know this angle because we Know this is a vertical angle to that one. How do we get this angle then? Oh, I I see that

there are corresponding angles here. Again, this pair of lines, it's pretty much guarantee you have to use corresponding angles somewhere, right? So, this is the same as this one. This is 120 because the corresponding angles and look at that supplementary angles, right? So, because of the supplementary meaning angles on the same line, these two angles have to add 180. So we can just do 180 minus 122 to get this angle which quick maths it's 58 I think. Yeah. 180 minus 122 is is 58. Sorry this is a mess. Let me let me clean this up.

Okay. Actually yeah let me let me let me just just start over. Okay. I'll do this really quick. So we have a triangle right? And we know we got this was 43 because vertical angles with angle A. And we got this one is 58. Yeah, cuz uh 180 minus 122 Should be 58. Okay, so we know these two angles and we get this one which is 180 minus these two. And by the way, like you should be using Desmos for pretty much every question. Let me let me put Desmos so I can just type type

this in because if you're using mental math, first of all, it's slower. Usually it takes more brain power if we like think about what the math equals. I would just use Desbus because it's usually faster. It prevents mistakes. Just just use Desmos every time. So 180 - 43 - 58 that's 79. Okay. So we know that this angle is 79. Then we can just get the last two because we know this is supplementary angles with these two, right? Or we just get this this one divided by two. So 180 or you can also do like

180 equals 79 plus like the angle we're trying to find and then do a regression. If you don't know what I'm talking about then you should definitely watch my Desmos Guide video. But the form of this angle plus this angle 180 or you could also do 180 minus 79 to get the same answer. It doesn't matter. So we know that this big angle equals 101°. This is 101°. We know these are the same. So we just divide it by two, right? So that's just 50.5, right? 101 101 / two is just 50.5. So we know

then that W equals 50.5. Hopefully that makes Sense. And these questions become really easy as you do more of these. But again, the general idea is get triangles and then take advantage of the triangles because you should know like in your head like, oh yeah, so I just need to find two of the angles. Let me how do I get that? And you you work backwards like that. Okay, now let's do another question. Let's do question 18 here. So I would try try to pause the video and solve it yourself and then see how I

Solve it. Okay, so there's two is triangles are shown. So right away it tells us is also like does the the length. So we know this is the same as that because this there's the same amount of slashes for this length. So there's one slash one slash. So these are the same. These are two slashes, two slashes. So these are the same as well. And these two line or yeah lengths are the same. these two links are the same if 180 minus Z. So that's just the Y and Z here because it just tells us

the relationship. So basically I'm pretty sure this tells us that if we knew what Z is, we can get Y. We use this formula to get Y or vice versa. If we know Y, we can get Z using this formula. And you can't assume that these are the same because these aren't vertical angles. Vertical angles is when straight lines intersect each other. Okay, these aren't straight lines. This is a crooked line. This is like a like that line, right? This. So those aren't straight lines. You can just tell just by looking at it. It's pretty

obvious. Okay. So you can't use vertical angles by the way. Y and Z aren't the same. So we have to use this formula to get the get these. And we know why. Okay. So it tells us why. So we can get Z immediately. And we need to find X. Okay, that's easy. So isoclesles triangles. So we just need to get one of them. We already have one of them, right? It tells us why. We can use that To plug in this formula to get Z. and then we can get these two angles and then do

supplementary angles with this one to get X. Okay, that hopefully that strategy. So you take advantage of triangles basically. That's that's really the strategy. Get like first first look at what you need to find. See how that relates to a triangle like this relates to this angle or how do I get this angle? I just need one of the cuz it's also a triangle. I just need one of The angles and can get the angle by getting that. Okay. It's just a mixture of those. Okay. So let's just so 180 minus this was the formula,

right? 180 minus Z equals 2. Yeah. Equals 2 Y. Okay. So we were given y. I think it said y= 75. Yeah, y= 75. Let's just type in 75 here and just see. Okay, so z equ= 30 then. And again, if you don't know this is regression. So like this is you should learn this. This is Desmos video. It's Really simple. This is how tells you to estimate the the unknown constants and this is going to give you what Z is. Okay, but I would recommend you watch the Desmos guide if you haven't. So this

30° and then we can get the other two here cuz we know this one. We can get these two which is just 180 minus 30 and then the ones that are left would just be these two and these have to add up to 150. So we just divide 150 by two to get what the angles are. So 75. So we know This is 75. Then we can also this one 75 too. But this one's 75. We just do 180 - 75 to get the remaining angle. So 180 minus 75 because supplementary angles, right? So that

must mean that x= 105 because of that. Okay, hopefully those questions should seem easy to you now. Like it's pretty simple. Okay, and you can do practice on on these all you want too. Okay, now last thing. I didn't this isn't one of the seven patterns. I've only seen one question like this on The history of the digital SAT that asked that required you to know this, but I'm just going to show it just in case. I think it was exper it was an experimental question because they asked about it in 2024 and they haven't

asked a single question that has to do with this rule again. But this is the rule for the sum of the interior angles of a polygon. Okay? And the formula you have to know for this is 180 * n minus 2. And n is the amount of sides for a polygon. So a polygon is just a shape. Okay? Like a polygon is just a shape of sides really. And if you if you seen the shape the measures of the inside angles of the sum of the inside angles. So let me let me give you an

example. So for example, here's like a a random shape. Sure. Okay. The this is not even straight lines. Okay. But then I I drew like a nice looking shape. So the angles of all these when you add Them up it should follow this formula. So for this it has one two three four five six yeah six sides. So if we just plug in n is how many sides. If you just plug in six for this that means all the angles in here should add to 720 degrees. Okay. And by the way, you don't need to

memorize this formula because you can just like get it yourself because we know a triangle has three sides. It has there has to be 180 degrees. We know a Square has four sides and it has to be 360. Hopefully you know a square or like a rectangle has to add up like the angles have to add to 360. Okay, if you didn't know, now you know. You can just get this formula again which is exact same. Like this is the exact same thing as 180 times xus 2 which is the same thing that we got.

EC it's the it's the exact same right so you can also just get the angle get this equation yourself if you forget It it's like oh so yeah a triangle has add up to this many rectangle has add up to this many we have two points on this line let's just get the line and you can just get the equation yourself okay anyways enough of that so this question should be easy for you now because you you know the formula right so try solving it okay a polygon has exactly 99 sides sides. So we just

plug in 99. Okay. And then what is the sum of the Interior angles? This is literally what this number is. Okay. This this equation gives you the sum of the interior angles is the formula for the sum of the interior angles. So 70,460 is the answer here. Okay. Yeah. And this this is the formula. You can memorize this if this didn't make sense to you, but I feel like you it should make sense to you. like I I would just rewatch that portion until it makes sense to you because it's annoying to just memorize Vague

formulas. But if you understand like where the formula comes from and like how to get it, then it's much easier to memorize it. So I would try to like figure out what I did here. So watch the Desmos course obviously the Desmos guide if you haven't done so already because this would like make a lot more sense what I just did with the regression here. Okay, but that's really it. That's angles, lines, lines, triangles, which is actually a decent Part of the the SAT if you look at it, at least in module one. It's a

decent part. Now, let's move on to right triangles. For right triangles on the SAT, all you need to know is these four rules, which I go over one by one. And something that students get confused a lot is these four rules only applies to right triangles. You can't use SOA, you can't use Pythagorean theorem on triangles that aren't right triangles. And right triangles are just any Triangle that has a right angle in it. Okay? So the question should give away that it's a right triangle because it says it's a right angle or something. For example,

here's a triangle. Um this symbol usually denotes a a right angle. So if you see like a square at at the bottom of an angle like on an angle that means it's a right triangle. Okay? You think like a square or right like 90 degree angles, right? So right angle. um it should say it's like a there's a Right angle in it or it should say it's a right triangle or like you see this squares looking thing on one of the angles then you know it's a right triangle. If it doesn't do that then you

can't use these four at all. Okay, you have to use what we talked about earlier with the isosles like scaling which is you just need to find one angle to get the other two angles things like that. Okay, you only use those. You can't use this stuff. So now let's talk about SOA. So this should be a refresher for most of you. But I I'll go over this. Okay. So sooa what is sookoa? So sooa is an acronym. Okay. First of all you need to know the different trig stuff. So s cosine and tangent. Okay.

And better I guess better known as sin n. Um because this is like on the calculator or like on the SAT you'll see like sin. So it's sin of like an angle. Let's just use x and like cosine of an angle and Tangent of an angle. Okay. So you see this is if you see co tan or or or sin n then that's s cosine tangent. Okay. And sooa is an acronym for how to get these values or if you have these values how to get the sides vice versa. Right? So let's just go over

this. So soul is just sign uh is opposite over hypotenuse. That's what it's saying. Okay? And cosine is adjacent over hypotenuse. And I'll go This more in depth. I just want to show you. So tangent is uh opposite over adjacent. Right? So you just sound it out. So O, right? So s equals opposite over hypotenuse. Okay? What I mean by like opposite adjacent hypotenuse is a right triangle. So let's say we have an angle. It's one of these two. We didn't know this is 90°. Okay? So let's say we were given one of the let's

say we're given this angle. Okay? We're given an angle X here. Opposite to this angle is this one. Right? This is the furthest length or furthest side of this triangle from this angles. That's the opposite angle. This is the adjacent. Adjacent means next to. Right? So this this side is next to adjacent to this angle. That's adjacent. And hypotenuse is just the longest side of the triangle. So whichever the triangle is, it's the one across from the 90 degree angle by the way. But it's just the the side longest from well the Longest side in

that triangle is hypotenuse. Okay. So basically what this is telling us is sign. So s sign s is going to be equal to opposite over hypotenuse. So let's say we're given sine of x. So we're given sine of x then and we were told let me actually put side so you you can like follow along right so let me put just like random side lines for all this let's do 345 that's like probably the most common triangle length you'll see okay let's Say a question just straight up asked us what is sine of x okay

so what is s of x then we can just do s oh oh sooa so so right so s o h so s equals opposite hypotenuse. The opposite side of this is four. Hypotenuse is five. So s= 4 over 5. Opposite over hypotenuse. Okay, that's how you can do this. So you need to memorize soa. Let's do another one. Let's say cosine of x. What is cosine of x? Cosine of x is is adjacent. So co so It's a c, right? C is a like cosine equals adjacent over hypotenuse, right? So adjacent it would be

three. The one next to it is three adjacent to it and hypotenus is five. So 3 over 5. Okay? So cosine of x would be 3 fths, right? And then tangent of x. What is tangent? So toa toa t equals o like o to o right? So you could do three uh opposite. So opposite my opposite over adjacent right. So the opposite to that angle is four. So and adjacent to that angle is Three. So 43. So tangent of x would be 43. Okay. Hopefully this makes sense how you can take advantage of sooa. Hopefully

that was more of a refresher to you. But um that this should explain what satoa means. You can again you can only use this for right triangle. But it's just like let's say we need to find this angle. We're given the sides. We can we can use that. You can see, oh, this sine of x must equal this and just Have the calculator solve for x for us. Um, or like I just show you real quick because I feel like people are going to get confused if I say something and I don't actually do it.

So, let's say sine of x is equal to 4 fths, right? The calculator can find x for us right there. Oh, shoot. Yeah, the calculator can find x for us. Or let's say it's um we're given the angle and we need to find like what what this is, right? cuz we know we know the hypotenus is five But we don't know what the opposite side is. It can do that for us too like it tells us. Okay, enough of that. Let's just go back here and let's talk about pag and fume. Pag and fume is

just this formula. It's literally in there. This is the SAT reference sheet by the way. It's literally in there. So on the the SED testing software, you can press reference on the top right of your screen when you're taking the test and You can see this this picture. Okay? And you can see oh C^²= A2 + B². Just to show you what I mean, let's type in the formula. So again, the reference sheet said C ^2 = A2 + B 2. And let's say we're given a triangle. Okay? And let's do a triangle, a right

triangle, by the It has to be a right triangle. And like this is a right triangle cuz it like told us this maybe. I don't know. Okay. And we are given three And four. And we need to find what this is. That's easy. It's a right triangle. How do we get we know two? It's just like a regular triangle, right? We need if we know two of the angles, we can get the the other one. Same thing for a right triangle for sides. If we know two sides, we can get the third one by using

Pagan theorem. So um oh how do we get this? So C square by the way C square if you look at the reference sheet C is Hypotenuse. Okay C is going to be the hypotenuse. A and B it doesn't really matter which one you pick but this is the other sides the other the other legs of the triangle and C is hypotenuse. Okay so if we go to our example if we know A, let me see if I can type. So if we know A, A, let's say A is three and B is four. It's

just it doesn't really matter. is just like the other sides and then we can get C, right? Just have Desma solve for C for us. Five. Answer Is five. And that's pretty simple. And we can do vice versa. Let's say we know um we know this one. We know five, but we don't know what this is. How do we get this? The same thing. Just use the formula again. Okay. So, we just plug in the the length. We do know we we know this is five. We don't know what this one is. Desmos can solve

that for us as well. It's three. Okay, hopefully that's making sense. Prog is is pretty simple. It's really in the reference G2. You technically don't even need to memorize it. Although I would cuz like at least you need to know the idea. If I have two of the sides, I can get the third one. Let's do a quick practice question just to confirm that you've understood what we talked about so far regarding triangles. So, pause this video, see if you can solve this question or at least figure out how you're going to solve the question.

You not actually have to solve It, but just like get an idea of how you're going to solve it. Hopefully you did that. So let's do it. So triangle PQR has right angle Q. So right away I I I'm confirming myself. Okay, this is a right triangle because it has a right angle in it. Let's draw this. By the way, you should always draw out any thing you can like in geometry and trigonometry questions. If you can draw it out, draw it out. Okay, so because you don't want to do it in your head and

Like mess up the angles in your head, it it's not good. It's just like extra few seconds to draw out. So we know this is the right angle. It doesn't really matter what the other ones are. Let's let's do P and and R. Okay. We just know it's a triangle and Q is the right angle. Okay, that's it. It doesn't matter what these are. Um like like R could be here, P could be here. Yeah. Okay. And then uh if sine of R equals 4 fths and like we see sign. By the way, The SAT

never test you on law of sign, law of cosine. If you don't know those all, then don't worry about it because SAT doesn't test you on them. They don't even they don't even test you on trig identities, which you probably learn in pre-calc. Um I take that back. They they do test you on trigger identities but you don't need to know them because you can just Desmos them like I have seen I think it might have been experimental question but I have seen triggered Identity questions like for example like sign s of something like 30

squared plus like cosine of something square I don't know like like 80 squar I know I messed up the square at different points but you have ideas like they test you on like questions like this and then they ask you oh what is this equal to and give you answer choices or you type this in. Look at what the what it's equal to. You can convert it. It doesn't matter, but you can convert it to Degrees if you want to because it's in degrees or something. Okay? But like you just type it into Desmos and

then just type in the answer choices and see which one's the same. Like you don't need to actually know it. Or if they ask you what is the value of this, just type it in. Desmos tells you. Okay? You don't need to know to get identities. Okay? There's a short sidrack. You should watch my Desmos guide video if you like want to learn more about um just like Typing stuff into Desmos, I guess. Okay. So, let's solve this question. So, by the way, if you see trig, this is a dead giveaway. That's a right triangle

because they won't test you. They don't use trig stuff unless it it's it is a right triangle. Like, you won't see a trig question involving stuff that's not right triangles. So, sine of r equals 4 fths sa s o h. So, s equals opposite of hypotenuse. The opposite angle to r is is the the PQ line here and this length. So we know opposite over here. So four this would be four. Um I yeah you can just type in four here. I I'll show you later with like other stuff you can do because it's really

this is a proportion not really like the exact side lengths but I'll talk more about that after the question. And hypotenuse is five. You could do this. Okay. Now what is the value of tangent of P? So tangent of P right Tangent is is opposite over adjacent toa right so we need to get this over that the problem is we don't know this but luckily we just learned the pythagorean theorem so we can get this because we know two is so we use pagan theorem if you type this into desmos it would be three by

the way so if you did 5 squar and then I don't know four squar you plus like what you're trying to find right it's three. Okay. So if we know this is three basically. Okay. And then We could do opposite. So by using the you can find this three and then you could do opposite over adjacent. So 34 is the answer then. Okay. But um this works perfectly. I just want you to understand though like be careful that this is just proportions. Let me erase this. It's so short. It's a slightly better delay. Okay. So

the these are proportions. Okay. Four fifths. We just know that this is four like four this side over this side equals four fths. That's all we know. Okay, this could actually like this side could actually be like like eight for example. Not 48 but just eight, right? And this side could be 10 for example. And then when you do 8 over 10, it's still four fifths, right? Because if you simplify the fraction 8 over 10, it's 4 fths. So all we know is that the fraction of this over this equals four fths. We don't know

for sure that this is four and this is five. Okay, that makes sense. We just Know that this is like if you put this over this is equals 4 fths. Like it could be eight and 10 for it could be like 16 and 20 for example. Okay, so I just want you to understand that. But we're fine because this is also proportions. So like proportions don't don't really matter. Um I'll like but like a way you could just like make sure that you're doing it correctly is let's say this is the right angle again. This

is R. Um instead of four, you put 4x, Right? Cuz it's just like and then and then put 5x here. And this just ensures that the proportion is the same. So like anytime I like as long as 4x I don't know what x is but as long as you multiply like some number to four and you multiply the exact same number to five. If you do four four times that number over five times this number is always going to be equal to four four fths because like the x will cancel out and like um hopefully

this isn't over Complicating it but I just want you to understand like the nuance of this. This is a proportion right the is the proportion of this to this is four fifths. That's what it's saying. It's not saying that this is actually four. I just want you to understand that in case that causes some complexity. I don't even know what like how that could cause complexities in the question. But this might like open your your mind a bit to just help you better understand trick Stuff. But like you could do 4x and then and then

you could just solve for this one. This would be 3x and you can literally just type in the the exact same thing. I don't know like uh uh like five instead of instead of doing all this you could just do 5x um 4x oh shoot sorry it's the square is supposed to be after the x and then um some mystery you don't know what this is but you know has to be x something right So uh we don't know if it's actually true or not so let's say like a x for example and then you

can have desmos solve for a for you which you're going to have to like yeah you should watch my Desmos uh guide if you don't see what I'm doing but it tells you it's three so you could do this too if you want to like this is being really nitpicky hopefully you understand this is a proportion okay it's not like saying that's actually Four next is special right triangles and this is also in the reference sheet it's right here so basically special rate triangles there's two of them okay there's a 30 60 90 triangle and

a 4545 90 triangle and when I say 3090 I'm refing into the angles of the triangle. So if you have a triangle that has the angles of 30, 60, and 90, then the the sides of the triangle should follow this relationship. So basically, if we knew what the side length of the side that's Across from the 30° angle was, so the 30° angle, the side across from that one, if you knew what this one is, then we know the hypotenuse has to be two times greater than that, that that length. and the other the side

across the 60° angle has to be that length times root of 3. Okay? So for example, let's say we knew x was equal to 1. That was one. Then we know 2x, right? If x equals 1, then 2x would be two because 2 * x, which is equal to 1. So 2 * 1 Equals 2, right? And if we knew x is 1, then 1 * 3 would just be 3. So we know this is 3. So that if you just know 369 triangle is just like the relationship before right when we went over the

the angles lames triangles and where for a regular triangle you just needed to know two of the angles right but for isocles triangle you know one angle it's just like this okay so for a regular right triangle you you need to know two of the side lengths to Get the other one but for a special right triangle. You just need to know one of the side lengths and you can get the other two. Hopefully that that makes sense. It's just like it's triangle but it's for side lengths. Now I that helps you. I'm just trying

to draw analogy so you can better understand this. Okay. And a really common pattern for 30 609 triangles is if you see a a equilateral triangle, right? So an equilateral Triangle is just all the angles are the same. So all the angles are 60° and it's biseected. So like it's cut in half. Then this triangle is a 369 triangle. This triangle is 369. So you think about like a equilateral triangle, right? There's 60° on each one. Right? So if I if I cut right down the middle, the 60° turns into 30°. So with this triangle

now, this is 30°. It's still 60°. Obviously 309 triangle, right? So this is just a Common pattern. If you see equalized triangle, you cut it right down the middle. that's going to be a 30 609 triangle that you see that all the time. Also, um a dead giveaway for 30 609 triangles is anytime you see a square root of three, like if if you see a square root of three in the answer choices or in the question, anywhere you see a square root of three, it's it's like almost guaranteed that you have to use a

3090 triangle some of three right There, you probably have to use a 3090 triangle somewhere. Okay, so that's a dead giveaway, too. It can help speed things up because you can limit down like oh the amount of things you have to like check to like figure out how to solve the question. You immedately know oh use 369 triangle let me just see like where 609 triangle could be and then just go from there. It's much easier that way. Okay. The other special triangle which isn't as common but still Pretty common is a 45 45 90

triangle. And the most common thing you'll see from this is like first of all if you see a square of two like somewhere in the question that's a pretty dead giveaway that you're going to have to use a 45 foot 90 triangle at least like like 98% of the time you see a square root of two anywhere in the question. Also if you see like a a square pretend this is a a square and you cut the square diagonally in half then what's Left is a 45 45 95 angle. So that's pretty it's just like

an equilateral triangle you split in half. So you might see this a lot actually. Um yeah because this is 90 degrees and this was 90 degrees but we cut in half now it's 45 degrees. If you also see another another thing that you might see is a isoclesles right triangle. Okay if you hear that or something to do like it's isoclesles right triangle or it's like a right triangle with two the silence are the Same then you know automatically it's a 45 45 90 triangle because what a 45 90 triangle is is just isoclesles right

triangle that's it. So like it's a right triangle 90° and we know the other side the other angles are the same because it'sles triangle and so it's by default we like these are 45°. So anytime you see a a isoclesles right triangle you see a square root of two if you see a square that's been cut in half diagonally it's a 4540 90 angle. Just Want to like help you recognize patterns. Okay but it's the same idea behind this right. So if we know like this side like any of the side these are the same

anyways but let's say this is one that automatically know that this one's one as well and we know the hypotenuse is 1 * 2 because this is s *2 okay it's the same thing as this okay so this thing for specialized triangles you just need to know one of the the sides so let's practice these let's do another Example question so pause this video see if you can solve this question on your own or at least figure out how to solve it okay I'm going to solve this question now so the perimeter of equil triangle

equilateral triangle. Right away I'm thinking 30 60 90 triangle right because this is just the pattern. If you see equilateral triangle there's a good chance you have to split in half and then there's two 309 triangles after you split in half like I mentioned earlier Um is 624 means the height of this yeah like dead giveaway. So you can see like those patterns on the SAT you see 3 triangle that should be 369 triangle. So we should make a 369 triangle pretty much to solve this question where K is a constant. What is the value

of K? Okay, pretty easy. So let's draw it out. always draw it out just to make sure there's no silly mistakes. Um, and then equalized triangle. So 6 624 each of that's the perimeter. So all the sides When you add them up that equals 624. Since all the sides are equal to each other, we just divide that number by three to get each of the sides. So 624id 3 is 28. I wouldn't recommend you do mental math in your head. I would like do Desmos. So I do 624 divided by 3 just to make sure

because I'm the type of guy that do that and I actually recommend you do that too. This is really it's like a few seconds doesn't you should already have decimals Like the calculator pulled up on your screen when you're doing this during the test anyways. Okay. Um 208 all these are 28. We should split it in half to get the 369 triangle because we already know this is three. So we should probably use the 369 triangle somewhere. Okay. Uh oh well that's like sticking now but okay. So this is going to be a 60° angle

because like the equal angle this each of the angles is 60° and we split this 60° in half. So this is just 30. Now That's the 309, right? 30 60 90. Okay. So 30 609 triangles. How can we use special rectangles for this? You see, let me clear this up. This is uh on the reference sheet and 2x which is what we had the hypotenuse, right? this the angle the side across from the 90° angle that's the hypotenuse right hypotenuse so we know basically 208 is equal to 2x right this is equal to 2x and

how do we get x then we just divided 28 by two which is 104 so x is going to be 104 Which you could have already gotten because we already like we know this side length is 28 we bisect it we did we we cut it in half to create the 369 triangle so this should be in half so it should just be 28 divid by 2 that works too. Okay, but thing is we need to find what the height of this triangle is to get k3. So how do we get the height? Remember if

we know x x is 104 the height is just this right x uh if you look at the the side across from the 60° angle that's the height right 60° angle that's that's x3. So this is going to be the height. If we know x is 104 we just multiply by 3 to get this length right. So it's it's just 104 * 3. And if you look at the question, let me erase that portion. If you look at the question, what is K? Like we K square3 is the height. Then you find K. So K

is just 104. It's like a number times of 3 is going to be the height. We have the number it multiply by this is the number That you multiply 3 by 104, right? So 104. I don't know why you wasted it, but you get the idea. So K equ= 104. Hopefully that that makes sense to you. The last thing you should know to master right triangles is the complimentary rule. And you actually don't really need need to know the complimentary rule. Most of the time you can just solve any question involving this with Desmos anyways.

But I I just want you to recognize this pattern in case it does Appeal and you you forgot how to use with Desmos or you just don't know how to solve with Desmos. Uh then I would just like use the complimentary rule. Okay, the complimentary rule basically you can recognize this anytime you see like 90 minus X. If you see that it's a dead giveaway is a complimentary angle question. Basically the idea behind this is a sign of an angle is equal to cosine of 90 minus that angle. Let me just say the same thing

for the switch around. So Cosine of that angle is equal to sign of 90 minus that angle. Okay, you could just memorize this. I I again I'm kind of against like blind memorization. I would like you to understand exactly why the this like where this comes from to help you memorize this better. So, I'll just show you real quick. This is actually pretty obvious why this is the case. Okay, here's a right triangle, a right triangle. Let's use the 345 triangle again. So, I'm just going to Label side random side 345. I just want to

show you this rule in action. So, uh we know let's say this is the 90° angle. Okay. Um let's say this is X. What would be this angle if this is X? This is 90 minus x. Like how do we So if we know two of the angles, we can just 180 minus these two angles to get the rest of them. And since we know this is 90°, 180 - 90 is 90, right? So these two angles have to add up to 90. We already Have this angle. To get this angle, you just do 90

- x. Hope it makes sense. Like these two angles have to add to 90°, right? So if you know this angle, you 90 minus that angle to get the the last angle, right? So that's where the 90 - x comes from. Sorry, that's a bit big. That's a 90 minus x. Okay, now let's do this. So sine of x, what is s of x? Okay, sine of x again opposite over hypotenuse. Sin of x is 3 fifths here, Right? Not s of x. Opposite of hypotenuse. What is cosine of 90 - x? Okay, this angle

now, right? 90 - x. It's adjacent over hypotenuse. That's also 35ths. So this is why this rule is always going to work. Okay, it has to do with right triangles. So b b b b b b b b b b b b b b b b b b b b basically this is just like this rule but in like you don't have to think it's like in the formula, right? So same thing for cosine of x cosine x 4 over 5, right? Adjacent Of hypotenuse. What is s of 90 - 90 - x the one

across opposite of it is four and the hypotenus is five. So it's still four fths. So you notice uh like this rule works because this is how like triangles work and like how trigonometry works but it's just in this form. So anytime you see a 90 - x you you can immediately know okay you probably have to like if it asks you what is what is cosine of x and like it gives you this you just oh it's it's it's literally Just the same value like it tells you what this equals you just lally that's

the exact same answer. Okay. So, it's just things like that. And I'll show you a quick example question. I also show you like how you can solve questions like that with Desmos. Um, as as you just don't overthink it. It's pretty easy. Okay. Oh, it's for Okay. Anyways, um I feel like I accidentally spoiled this question, but try this question. See if You can solve it. So, it says one angle measure x where x Yeah. So, it tells the sign of x and what is cosine of 90 minus that? So, the answer should just be

four fths. Hopefully that makes sense to like it's literally just the same like s if you look at the formula again sine of x equals cosine 9 - x it gave us s of x and we know that that sin of x has to equal cosine 9 - x obviously cosine 90 - x is just the same number like it is the same number right so the answer is lally Just four fths because that's that's what the that's what s of x equals okay but you can also solve this with desmos actually if we pull

if we do So sin sine of x sin of x right equals 4 fths. We can get what the if this is in degrees. You can work with radians too if you want to. It doesn't really matter but that's in degrees. So we have x and we can do cosine of 90 minus x and we'll get the answer which Is 4 fths. But you can solve a desmos too. You don't need to know how to do like I literally just had Desmos solve for x for me and I just plug it in and I got

I got the value. Like it's pretty simple. This is a standard just like single variable equation. You can watch my Desmos course like on on my YouTube video like I'll have it somewhere on the screen. Every single time I mention that I just like have it so you could you can watch it. You can click on that to watch It if you haven't watched it already for some reason. Okay, but that's right triangles. They're pretty simple. Just remember that these four rules can only apply to right triangles. Okay, it's mainly just these. This is kind

of rare that you'll see this complimentary rule, but mainly just these. And like you use these a lot. Make sure you know them. You can recognize them. And anytime you see a right triangle, you can pretty much just do a checklist of these three Things. You can go like, "Oh, is is there soa can I use toa heal somewhere? Oh, no. Can I use pagan and f?" If no, then it's probably going to be third one because this is like it has to be one of these or maybe a combination of these three. But it

helps you narrow down what you need to look for. Okay? Because you you're not like the person that tries everything and spend five minutes and still can't figure out what the question. you know exactly what the SAT Is going to test you on, you can just do a checklist to see which one applies. Now, let's talk about similar shapes. And if two shapes are similar, it just means that they're the exact same, but one of them is bigger than the other one. So, the most common similar shape question on the SAT is is triangles. Triang

they love triangles on the SAT. Okay? But, uh, so let me draw two triangles just to show you. So if I have these and if I said these triangles are Similar, that would mean that they're the exact same, but one of them is just stretched out more like bigger than than the other one. For example, if I made like three for I'm just going to draw some of the shapes for you. So okay, so I said these the the sides for these is 3 4 5 and the sides for this is 6 8 10 and

these are similar shapes. Okay, you should be able to notice something pretty obvious about these and it's that every single side of this triangle and You look at the exact same side on the other triangle. The the bigger triangle is always two times bigger than the smaller triangle, right? You multiply by two to get from this to this, right? The this side like the the left most side, right? You multiply by two to get there from from this triangle to the other one. Same thing for the bottom. You multiply by two from this one to

get to this one. Same thing for this one. multiply by two to five to get to 10. So It's always multiplying by two. And this is what similar shapes means. Okay? It doesn't have to be two, but it's just you have to multiply by the same number from each side to get to the other side. And also similar shapes also have share the same proportions as well, but between the the shapes between the sides of the shape themselves. So for example, notice this. If you make a fraction out of this, that's 6 over 8, which

is like this over this, like the I guess the two Two legs of this triangle. 6 over 8, that equals 3/4s, right? And if you do if you do the exact same thing, these two, it makes 3/4s as well. And same thing if I did 10 over 8, that's going to equal 5/4s as well. So it's similar shapes, basically exact same shape. Okay? Like everything is the same. The only thing that's different is the sides are stretched out more in equal proportion between all the sides. That's what similar shapes means. And just to Help you

understand this more, squares are always similar shapes. Like if I have two squares, they're by default similar shapes because a square definition of a square is all the sides are the same like in in a foursided shape, right? So if I know pretend that's a square, I know that's actually a rectangle, but then all this all the sides would be this is one. Then if this is a square and I know like all the sides are two because it's a square so All the sides are the same. These are similar because if I multiply a

number from this to to this side length to get to this side length it's going to be exact same as multiplying a number from this side like it's going to be the same like the scale it's called the scale factor like how how big the other shape is compared to the other one. Like the scale like what number you need to multiply to this to the smaller shape to get to the bigger shape. That's the Scale factor. Okay. But like there's a consistent scale factor basically between all the sides. So, and by the way, it

could be 3D as well. It could be cubes. It could be like a rectangular prism. It could be pyramids. It doesn't it doesn't just have to be 2D. So, similar shapes have to have the same proportion between all the sides. And also another way you can prove that shapes are similar is all the angles are the same as like the corresponding Angles like the the top angle here is the same as this angle. At least like for like uh like any shape all the there's at least like a matching angle for every single angle. Like

it's the exact same. So like if this angle is the same as this angle, this angle is same as this angle, this angle is the same as this angle. So all the angles there's a matching angle on the other shape, that means they're similar as well. So if all the angles are the same or the sides of Proportional, that means they're similar. And we'll go into proving triangles in in a bit after after I go over similar shapes and air and volume questions to help you like there'll be questions that ask you to prove similar

similarity as well. Okay, but let's go into like the most common type of similar shape question and we go to the like the harder more unusual ones. Like I mentioned before, the most common type of similar shape question have to do With triangles. And actually anytime you see more than one triangle or there's multiple triangles in the question, it's pretty much guaranteed that those triangles will be similar to each other. So anytime you see multiple triangles, it's a similar triangle question. You can just follow the standard process. So, there's a set four-step process I use

to solve every single similar triangle question. It's pretty much this works every single time and I'm I'm Going to teach it to you. So, first I'm going to solve this question in front of you and the next question I'll have you pause it and see if you can use this process for the other question. Okay, but first this question. So, uh some camp counselor wants to find length X, right? So, just this this thing the length. Okay. So, this gives us a bunch of lengths and it says um respectively segments like the B. Okay. E.

Okay. Sure. It says like these two angles have The same measurement which by the way how you read this cuz I didn't explain this. A E B is you just trace it, right? So A E B. So A E B. What angle did that create? It created the angle right there. So that's that's that's referring to angle E technically and C DB C DB. So that's talking about this signal. That's how you read that notation. Okay. So the first step of this process is to draw the triangles. Draw the triangles and make them look identical.

So what I mean By that is right now the corresponding angles aren't lined up. Like it's not really clear. I mean it's kind of like obvious which angles are the same to each other, but you want to draw them out in like a neat format, right? So you can actually like use it. So let's uh draw out two triangles. Make them easily big. Does not does not have to be perfect at all. Basically the idea is just to draw two triangles. That way you can label corresponding angles and Sides. Second step is to label the

angles and make sure you don't mess this up. But basically label the angles that are equal to each other on the same uh sides of the triangles. Right? So we know B is the same because this is vertical angles. Uh this is how like previous knowledge of in this video is going to come into play because you this it kind of builds off of this. So we learned about vertical angles. I guess like one of the first things we actually Was the first thing we learned about pretty much was vertical angles. So you need to

know like how to you you have to be comfortable with your your lines and angles to to solve similar triangle questions because you want to prove that these angles are actually the same. You can't just assume it is. I think like if you actually like you could just assume cuz they're pretty much always similar angles but it's always good to just like prove for sure that this angle is Actually the same as that angle or else you get the question wrong. Like I've seen a lot of students like they accidentally think that E is the

same as C and then they they line up the angles that way and then they get the question wrong. They have to make sure they're actually the same. That's actually a really common mistake students make. And also is we're told AE is the same as C DB. So let's just say let's put that in. So um if this is E, Then we know that D is the same as that. You see it's actually that this triangle is like this triangle but it's flipped upside down. And a lot of students miss that because they don't they

just like oh is easy but they don't actually take the time to draw it out and they were the angles. Okay. So that leaves us with A being the same as C, right? Okay, now it looks a bit but you get the idea. Okay, so this is the same. Next step is to label the known sockets like The lengths that we know. All right, so we know the 1800 is same as 80. So let's do this real quick. So 1,800 is AB. Um, 1400 is EB. This was just me reading and understanding the question. 1400

this is telling us which uh sides are which measure. So, uh, 700 is B D BD 700. And then lastly, CD is 800. So, it's eight. [gasps] Uh, that's a really bad Okay, let me zoom in. 800. Okay. Um, that's not as zero. Oh, well, I tried. So, 800. Okay. And what is the value back? So, x is a e. So, we need to find this measure a e. Okay, so what is the last step? Set up the fractions equal to each other and plug the single variable equation into Desmos. Let's get Desmos uh Desmos.

Okay, so remember what similar shapes is about? Like what does similar shapes mean? Uh let me I want to I want you to see this. It's like not Showing up. Okay, there we go. So what does similar shapes mean? The idea again is that all the proportions are the same. Meaning if I were to multiply a number from 700 to get to this number, which would be two, right? This is two times greater than that. 1,400 is two times greater than 700. I It's the same exact number from this to that, right? The scale factor

is the same. So what I like doing is getting solving for the scale factor. So what's the scale Factor? How big is each side compared to the corresponding side? It's two times. But how do you get that? It's just like this side divided by this like the big side divided by the small side. That's the scale factor or you can solve that's like 1400 divided by 700. So like this is how we can solve for like if we had this angle this this side we could do 1,800 divided by that side to get like how

how many times greater this side is to that side. Okay, this basic math. How Many times greater is the bigger triangle to the smaller triangle? Okay, that's the scale factor. And we just set the scale factors equal to each other because we know the scale factor has to be the same across all the sides. Right? So this side, so because this side over this side equal two, then that means this side over this side must also equal two. So we can literally just set this equal, right? This side over this side because we know they're

the same. Like The big triangle, the corresponding side of the big triangle over the corresponding side of the small triangle. It should be the same. That's how this works. That's what I mean by setting up fractions. You don't have to cross multiply. I just have a desol for you. Just make make the x a constant by putting one here and replace the equal sign of a squiggly line. Just the normal stuff. You should definitely watch my Desmos guide if you haven't already. It's like the fifth time mentioning that. Okay, so we know X is 1,600

and that's it. You can also do this in your head. Like I understand like it's two times greater. This one's a pretty easy question, but like this is two times greater. So you just do 800 * 2 it's 1,600. That that works too. Now let's go to a decently harder question and I want you to solve this question on your own. So pause the video and try solving it. Now there are two ways to solve this Question by the way. There's area using the area of a triangle. You can use that to solve this question.

There's also similar triangles, but I want you to use similar triangles. So, the exact process here, this process, solve it using this process. Okay, hopefully you pause the video and tried solving it. Now, um this is actually a really common shape. You see this shape a ton on the SAT, so you should be really comfortable with Working with this shape. Basically, it's like one big triangle, one big triangle. Let me just show you what I mean. So, this big triangle and inside a big triangle are two smaller triangles. Okay, if you see this shape, this

exact like right triangle, two triangles inside it shape, then okay, you can use similar triangles to solve this question every single time. And notice there's three triangles. So we can draw we have to draw three similar triangles. These Are all similar by the way. Let's draw them out. So uh uh here's the big triangle. Here's it doesn't really matter like you don't have to get it perfect at all, but I just want to draw out three triangles. Let's say let's say this is a big triangle. So the right angle is here, right? And these are

the right angles. So let's label the right angle. So N is the right angle for the big one. Q is the right angle for the smaller one. So Let's label Q and Q. Okay. And let's find the other one. So M. So if let's say this is the smallest one and if we say this is M, then obviously M is going to be the exact same as the big triangle, right? It's the literally the exact same angle. So we can say M is the same as the M of the small one. And P, same thing.

P is the for the the medium triangle like the second to smallest triangle. P is the same as the P of the big triangle. So we Can just say P is the same there. And that leaves us with the let me erase this so you can see it better. That leaves us with the angles that are left. All right. So that means that QP QPN. So N is left and this angle N must be equal to this M then. And also this N must is left here. Yeah. Q Q Q MNT Q MNT. Okay, cool.

Hopefully that makes sense how I did that. You notice I was I was making absolute I'm not just assuming that that these things Are the same. I'm actually proving to myself that these are the same because this is the literally the exact same as that one. Okay, I can draw that one there. That's same as that one. I can draw that one there. So make sure you actually prove to yourself that those angles are the same. Okay, next let's label the known sides. So we know MN is it tells us right there. MN is three.

So MN let's say put three here isn't yeah MN there's usually multiple MNS in this Shape. So MN3 NP is four NP 4 NP right there four. Okay so we know that we need to find the length of N cub. So we need to find this. Okay it's also here as well. And this this is this is good enough right because we have three four. We could solve for the other side right there. I this is this is a 345 triangle I already know. You could do Pagan and theorem. If you don't know Pythagorean Triplets

I wouldn't worry about it if you don't know Pagan triplets. You don't need it. This use and fume because the right triangle we know two sides. We can use Pythagorean and to get the third sign. This is five. Just doing that. Okay. So then we know okay three five. So we have enough right. Um we can get the the scale factor between this triangle and this triangle right this and this. So we just do like 54 for example to get the scale factor and then We just set 3 overx equal to that because the scale

factor should be the same between these two. So let's pull up desmos and just have this solve that for us. Pretty much no thinking involved. It should you should get to a point where like you have to think to do this but it's very standard. Right? So 54 that's the scale factor, right? And we can so we know that's a scale factor. We can just like use that to to solve for the other one 3 overx. Right? So the answer Is 2.4 for which is C. Yes. Answer choice C right there. Now, uh I just

want to make sure that you you don't accidentally do something like this. Like you don't put like X1 and then three here. Okay. It's not how that works. It has to be like if you put the the bigger length at the top here, you have to put the bigger length at the the top for the other fraction as well. Okay? Like the bigger triangles like the this is the big triangle. because we Used the the five from the big triangle up in the numerator for this fraction. So for the numerator for this fraction, we had

to use the side from the big triangle as well. So don't don't accidentally mess this up. And if you weren't able to get these three like you if you did not notice that there were three triangles just from looking at the shape. Now you know this is this is pattern recognition. Okay, this is actually a really common pattern. I need You to to learn this. So basically anytime you see this right triangle shape and there's like a like a it's by the way this is called the altitude. The height of a triangle is also called

the altitude. So if you see the word altitude which it makes sense but uh let me type it out so you can see al how do you spell altitude? I spell that I think there's x and t. I don't know that might be right. Altitude. Yeah. Okay. Altitude. So if you see the word out if You see this word that just means the height of a triangle. So if you see like for example oh there's a right triangle and then the the the altitude of this triangle bisects it by I don't know what the altitude

extends from here to here or some something like that. Okay it's going to be this shape. So if you see like altitude if you see like a right triangle it's just like another right triangle two right triangles inside that right triangle like that it's going to Be this shape. Okay it's going to be two triangles draw it out like this. If I had to solve the question exactly how I solved it. Here is a even harder question than the previous two questions. But this is also a similar triangle question. I want you to pause the

video, see if you can solve this. At least have get an idea of how you can get the answer and then hopefully you did it by now. Let's try doing this. So in the figure above, tangent B equals 3/4s. Tangent B this has to do with trig. So you need to know like what tangent means, right? So toa toa, right? Tangent is opposite over adjacent because the toa that's how that works, right? So three. So B this is um this three at least when you divide this side by this side it should be equal to

3/4s like this side by this side and also because like these are similar triangle these are two triangles in here right it's one Triangle another triangle these are two triangles right this the big triangle is the small triangles these are similar triangles they share like this angle they share this right angle and then obviously by default that means these angles are the same so these are similar triangles right all the angles are the same so this als also shows the proportion like three if I divide this by this it's going to be 3/4s as well.

So that's what tangent is. I guess this Like you could do like x2 like a different number basically. I don't know. Um yeah but like they should be proportional. So if you divide the opposite over the adjacent it should be three forms. That's what this means. Okay. It has a proportion basically between the two legs of the the right triangle. If BC equals 15. So we know BC = 15 and DA = 40. DA. So it's just this part. Okay. Interesting. So it's not giving us Like a straight length of the triangle. It's giving us

a small part. What is the value of DE? We need to find this. So how do we do this question? We do know one of the lengths and we know that it follows this like 34 proportion. So we can probably solve for X. Yeah, we can definitely solve for X cuz we know like we can do Pagan theorem even cuz I already know. Yeah, like this it should be like standard for this. I think that it's three like I can do in my head x= 3 Cuz I know 34 five triangles but um let me

show you like how uh like an average student would do this. Let me pick and so like this like you can see it's a lot harder because you have to use like a bunch of different knowledges from what we've talked about so far. Okay. So uh yeah sure I'll I'll deal with that later. Okay. So we know the hypotenuse of this big triangle is 15. Right. Okay. Okay. And then we know 3x sorry 3x this is 3x 3x Squar like a^2 + b²= c² right? So if we do 3x and we just treat that as

like a for example. So a square which is just 3x^2 plus b ^2 which is just uh 4x^2 and 4x. So this is b and I'm squaring this entire thing. Let me make sure to put the one there. It can solve for x for us which is three. Right? So this is how you can do it. Okay. As long as there's a single variable equation, there's only one variable you need desmos to find, it Can solve it for you. You just need to put in one equation. There's only one variable you need desmos to find,

it can solve for the the missing variable. If there's two variables, then it's probably not enough. You probably need two equations for that. But like we we've got in situation where there's one equation, there's only one variable that needs to find. So obviously you can solve for that that variable for us. Okay. um that actually comes into play a Lot for area in volume questions which we'll talk about in a bit. So uh we know x x at least we know x1 or like this this x which is three. So then we can get this

side length right 3 * 4 is 12. So we know this entire side length is 12 and then we can subtract it from four to get like this this this BD right. So 12 minus 4 which is just eight. Okay. And let me get uh what the heck happened to my Oh, there we go. Um I delete everything. Okay, so we know this is Eight, right? We because we knew x was three. We did 3 * 4 to get this this one, which is 12 minus the four here to get what's like this just this

side length. That's eight. Okay, so we know the proportions between these two sides. So like if this is 12 and this smaller triangle is eight, then we have the scale factor already. We can probably use that to get this, right? Cuz we know um again it was this one was three 3x. Yeah, 3x. This is four. Yeah. 4 * 12 to Get 12. Yeah. Four time. No, four * 3 to get 12. This is 3x. So 3 * 3 to get 9 cuz x equ= 3, right? So this this side equals 9. I put that

as big side equals 9. So we can we have enough. So we just do get the scale factor like this over this equals this over this. Let's just do that. Okay. Um so 12 12 over 8 is equal to 9 over x and the answer is six then that's where the e hold right. Hopefully that makes sense. And basically sometimes there's questions that give you the proportion and then you have to use the proportion to get the side lengths. And for this question, I didn't draw out the triangles like this. I mean, I could have,

but it it's already drawn out for me. Like, it's not like it's disorientated like like the previous questions. Like the previous questions were very disorientated. I mean, like it's like this one's flipped around. Um, This is like this is definitely like definitely flipped around. But for this triangle, it's already the corresponding sides and angles already lined up for us. So, that's why I didn't re you could have redraw if you want to, but I didn't. I just like found the the known sides and then used the proportion to to solve this question. >> Let's take

this up a notch and talk about a more advanced type of similar shape question. But these types of Questions won't be hard for you because you'll know something that I like to call the proportion trick. So what is the proportion trick? Okay, I'm going to use squares because they're similar shapes as I said earlier. Basically, you can only do this if they're similar shapes. Okay, like the question should say to similar or the question will like give away that similar because it says like the proportion between the sides are the same or something like that.

That's is what similar shapes mean. Okay, so let's let's first talk about finding the scale factor. Um let's say this was four and this was two. They're just squares, right? But so all the sides are are the same. So like this is all four. Uh this is all fours and then this is all twos. So what is the scale factor between these two shapes? And again the scale factor is what you multiply to the small shape to get to the big shape. Okay. So what do you need To multiply to like this two to get

to the corresponding side of the the bigger square. Okay. How you get that is you just divide the big by the small right? So so the corresponding side you take the big bigger one so four and you divide it by the smaller one. So two. So the scale factor is two. Right? So because you multiply two to two the small side length to get to the big side length. Okay, so that's the scale factor. Now, what about the scale factor Of the area? Because that's pretty simple. The scale factor of the sides. What about areas?

If all you know is this number, right? We just know that the scale factor of the silence is two. How do you get the scale factor of the area? Because it's actually not the same number. Let's say the area of the area really know this is four. The area of this is 16. You notice if I do 16 / 4, the scale factor is actually four, right? Okay, at least you when you do Areas, right? When you like what is the scale factor? What would I had to multiply the the area of the small square

to get to the big square? It multiply by four because 16 divided 4 is four. So you notice that's a different scale factor. Okay, let's try let's try something else. Okay, instead of instead of doing two and four, I see let me just erase everything and just redraw it. Okay, let let me do uh because I want you to like properly understand this. I Think this is like a good way to understand it. But let's do another thing. But now let's do like three and nine. Okay, so we know all the things. So that means

this area is 9 cuz 3 * 3 because the squares, right? 3 * 3 is 9. 9 * 9. This is this is going to be 81 then. And 9 * 9 is 80. I messed up. Uh 81. Okay. So what is the scale factor of the sides? 9id 3 it's three, right? So you multiply three to this side to get to This side. Scale factor is three. What is the scale factor of the areas? What do you need to multiply to nine to get to 81? It's nine, right? 81 divided by 9. 81 divided

by 9 is nine. So you need to multiply the area. So you're going from the scale factor of the sides is it's three. The scale factor is three. But the scale factor of the area is actually nine. So hopefully you're seeing a pattern. So before when the scale factor was two, The area scale factor is four. Now the when the scale factor is three, the the area scale factor is nine. So you notice the area's scale factor is always the the the side square factor squared. Hope that makes sense. So remember the scale factor of

the size before was two. If you square two that equals four, right? Two square is four. Same thing for this when the when the scale factor of the sides is three. To get to the scale factor of the area you square the The scale factor of the sides. So let's say all we knew, okay, we didn't know any of the side names of of the of these square. I guess you could figure it out because let's say let's say they were rectangles. Okay, there were two rectangles. There's two rectangles. Okay, so you like we just

know the question says they're similar, but we don't know the sidings. All we know about this is that the proportion of the sides to the scale Factor is five. That's all we know. Oh, and also we know that the area of this rectangle is 10. Okay, we know the area of the small rectangle is 10. We know these are similar and that the scale factor of the of the sides is five. Okay, how do we get the the area of the big rectangle? So since we know the scale factor of the size is five, we

can just square that number and then we know okay the scale factor of the area should be 25. That Means if you multiply the smaller area by 25, you get the bigger area. So this is the area of this big bigger rectangle should be 250. Hope that's making sense. Okay. So if you know that the scale factor of the sides, you can square that scale factor or the that proportion I guess the ratio you can square that to get to the the ratio or the scale factor of the of the areas. Now if you're working

with 3D so not just 2D but like volumes like not area but volumes we're Working with volumes. Let's say these were rectangular prisms and we knew that the scale factor of the sides was still five. But let me let me erase uh erase this one. Right? So this is actually going to change a bit. So, we know the scale factor of the sides is five. Okay. But we we now we're doing volumes and not um sorry, I need to change this up. Okay. So, now we're doing 3D shape. So, we're doing with volumes, not area

anymore. We know the Volume of this this rectangular prism is 10. What is the volume of this one? Again, we know the scale factor is five. And we know these are proportional. Um, I didn't teach you this, so you shouldn't have been able to answer, but if you already know this, then I guess you could. You might have been able to guess this. Okay, instead of squing the skill factor the side, you cube it. Now you cube it. So 5 cubed is 125, right? 5 * 5 is 25. 25 * 5 is 125. So we

know the Scale factor of the volume is 125. And then we do 10 * 125, which is just 1,250. And that's going to be the just 125* 10, right? That's going to be uh the volume of the the big rectangle, a rectangular prism. Okay, that makes makes sense. So, okay. So, from the scale factor of the sides, you square that scale factor to get the scale factor of the areas. So, you're going from area to area. If you're going from volume to volume, then You have to cube that scale factor of the sides. It has

to be the side. You can't you can't cube the scale factor of the area. Okay? Don't get that confused. You have to get the scale factor of the sides. Okay? the like the the onedimensional scale factor and then you can like go to threedimensional by cubing it when you go to the two dimensional by squaring it. Okay, I went pretty in depth into that. So hopefully like that's going to stick with you and You'll remember that cuz it like it makes really hard questions super easy if you just know that trick because most people don't

know that trick and you'll be way ahead of everyone that does. I mean that you'll be way ahead of everyone that doesn't know that trick. Okay, let's try this question. This is going to be on the easier side. I'll slowly make this question harder as we go on. So, pause the video, see if you can solve this question. Okay, let's do This real quick. So, the side length of square A B C D is twice the side length of square. So, right away, oh, similar shape to the squares, right? And it's twice. So, we know

the scale factor of the sides is two because it's twice the side length. Okay. Um, let me let me draw this out cuz I should, right? Um, so here's a square. I know it doesn't look look like a square, but we do our best. This is a b c d. I'm just going to Yeah. A B C D really quickly. And then the This is E F. Yeah. E F G H uh G H. Okay. Yeah. Why are E so annoying as well? Okay. Anyways, this is twice the size. So doesn't actually tell us any

of the side lengths, but it tells us the scale factor directly because we know this times uh wait is twice. So actually this one's the bigger one, right? So we get From this multiply side length by two for this to get the side length of the big one. Right? So this is the smaller one. This is the big one. If the area of square E FG so we we're given the area 9. What is the area of square A B C D? So how do we solve this question? It's simple, right? We we know the

proportions just like what we did before, right? We know the scale factor the proportion between this the the is nine. I'm just making sure. Okay. We Know the proportion between the two squares right of the sides. So if we want to go from the portion between the sides the proportion between the areas we square it we square the proportion. So now let's square the scale factors. Now we go two square which is four. So we know the proportion between the areas is four. So then we can do okay what is the the proportion? What is

the area of the big one? We just go from the small one multiply it by four. So 9 * 4 is 36. Okay, the area of the big the big square is going to be 36. That should be pretty simple. Now, let's do a harder question. Okay, I'm going to make this question a lot harder. Now, instead of the side length of square ABCD, let's do the side length of cube. Um, I'm I'm just going to say I'm going to keep ABC. You get the idea. It's like a cube, right? It's twice the side length

side lengths of cube a efgh. Okay. And if this The volume if the volume of square of cube this is nine what is the volume of cube this? Okay it's the exact same thing but now we're dealing with threedimensional volume instead of area. So what do you do to this? Okay this should be pretty similar in at least how you solve this question. Okay, but you just have to do one thing differently and this should be pretty obvious. Okay, so the answer is you just Instead of squaring the the nine, you you cube the nine,

right? Because now instead of 2D, we're working with threedimensional. So we I'm sorry this is like a bit too slow, but I'm like building up to a point. Okay, so let's say this was um this is the smaller one. Let's say this smaller one is nine. So we know the portion between the sides is two. We we cube this now to go from the side proportion of the side scale factor to The volume scale factor. 2 cubed is eight. So we just do 9* 8 72. Right? So the volume of the big cube is 72.

Okay. Now let's change up the question even more. And this is where it starts getting hard. Let's do surface area. Okay. So instead of volume, okay, it tells us the surface area. So it says if the surface surface area of cube this is nine. What is the surface area of the other cube? Okay, so we know the surface area of This is nine or surface area of this. Now the answer is is the exact same thing as we did previously with the area right this is just area and so we know if you know the

sides of the proportion is two we just square it and then we solve like like it's any other area question. So it's four, right? The portion between the areas is four because we squared the two. So now this the surface area of this cube is going to be four * 9 36. Okay, I think a lot Of you at least probably not most of you but a lot of you would have like multiplied by six or something because you think oh there's six sides and you multiply by six. It doesn't matter. Okay, as long as

it's area it's just it's a surface area. There might be a surface area but it's still area. It's two still two dimensional. As long as it's two dimensional then we all we do is just square the the proportion of the sides. That's it. Okay. And then we don't deal with area. If you're dealing with volume, then we cube it. Okay. So this is just even the surface area, it's still area. So you just square it um square the proportion to get the proportion between the surface areas. Okay. Now let's change up the question even more

and make it even harder. Let's say the side lengths. So instead of side lengths, let's do the um I want to change this back into volume. Actually, let's do this. So the Volume of cube this what is the volume volume of this but instead of side lengths I'm going to say the so area of cube abd is twice the surface area um surface area of efg instead of twice I'm going to do is 25 times the surface area. Just change up the numbers a bit. Okay. See if you can Solve this question now. So, I

changed up the question a decent bit. Hopefully, it's still okay. Um, I know it says like cube, but it's not a cube, but you get the idea. Okay. Try try solving this now. Okay. Hopefully, you like had a chance to to try to figure this out. This is probably as hard as a proportion trick can get. As long as you understand this, you'll be fine. So, the surface area of cube ABC D is 25, times. It's just saying the area of the big one is 25 times the area of the small one. Okay, so that's

the proportion between the areas, right? The volume. So we know this is nine and we need to get the volume of the big one. So we just need to figure out what the scale factor is between the volumes. Like what do you need to multiply to this to get to this one? The problem is I think a lot of you might have just like squared the 25 for some reason. That is not how you do this. Okay, remember how do we get the Scale factor of the volume? We have to cube the scale factor of

the sides. Okay, the scale factor of the sides. That's what we have to cube. So, how do we need to get the scale factor of the sides? Basically, how do we get the scale factor of the sides? We need to go from because we know the scale factor of the area and we know the area. We know the area is 25. I use the scale factor is and we know that this is equal to the side square factor squared. I'm put I'm just do S for the side square fac side side scale factor. Okay. So

we know 25 is equal to the side square factor Q the square is equal to that square. So we just need to square root the 25. That way it's going to give us the side of the the scale factor of the sides right. So we can just do 25 square rooted which is just five. So then we know okay the sides scale factor is is five and then we can cube the side scale factor to get the volume scale Factor. So five then that's the so we need to go back down to the sides. We

can only work between this the scale factor of the sides to get to the volume to the area. So now that we know factor scale factor of the sides we can cube it now. Now 5 cubed is 125. And now we know the scale factor of the volume. So I'm going to do 9 * 125 to get this number which I don't even know what 125* 9 is. Um my brain is not fast enough to do that. Actually it probably is but I'm Too lazy to do it. So I'm just going to do Desmos. So

125 * 9 1,125. Okay, that's the answer. That's going to be the volume with the the big one. Okay, hopefully that made sense. If that made sense to you, you you pretty much mastered the similar shape questions. This is as hard as it can get. I think there's a missing silent peel. Okay, you get the idea. Hopefully that that makes sense. And if that didn't make sense, I would replay This this last part. I just making sure because I've seen like they they have been asking a lot of similar There's been like one every other

test. So there's a decent chance you see questions like this and you should definitely understand similar triangle questions as well because those are super common like very common actually they're pretty much always at least one of them every single test. So make sure you know how to deal with similar Triangles and also um like scale factors and be comfortable working with like the volume and the area and like how to convert between the two and also like going from side length to volume things like that. Next, let's go over area and volume questions, which is

actually the most common type of question in geometry and trig. So, it's pretty important that you get good at this. Luckily, there is a SAT reference sheet that you get access to during test that pretty much Gives you all the formulas you need to know for area and volume. Now, however, I would recommend you actually understand where these formulas come from, at least for the simple shape. You don't need to memorize like where the circle I don't even know like I kind of know but like where where does pi r square where 2 pi r

come from I don't think you need to memorize that but I would like you to memorize not memorize but like understand where length* width Comes from where length* width time height comes from where like this this formula comes from okay so to help you understand this what is a rectangle a rectangle really is just a bunch of lines stack on top of each other if you really think about it like we just take we just take one length of a rectangle, we just stack it on top of each other until it becomes a rectangle. And

we just stack a bunch of lengths on top of each other. And how many lengths do we Stack on on top of each other to get this size of a rectangle? We stack with amount of lengths, right? You see like how many lengths we just stack, how many like however long it takes for you to reach the width amount of the width, right? Like the whatever this number is, whatever like this this measure of distance is. So that's where length times width comes from. It's just here's the length. How many lengths are there? There's width

amount of lengths. So this Is length times width. That's where the area of a rectangle comes from. Same thing like you can go this way too. Like horses vertically, right? But I guess we're going horizontally this way now. But basically like how many whiffs are there? There's length amount of whiffs. You really just like line up the whiffs side by side until you get a rectangle. And then how big the rectangle is is just like how many widths you stack next to each other, which is just the length. Okay, that's where length times width comes

from. Similarly, if you look at a rectangular prism, a rectangular prism is really just a rectangle like the base of this like the rectangle, but you stack up a bunch of different rectangles. Like here's another rectangle, here's another rectangle. You just keep on stacking up rectangles until you have a 3D shape, a rectangular prism. And that's where this comes from. So it's actually just the base of this Rectangle with length times width, right? Len times width is just this rectangle. So the bottom rectangle and you just keep on stacking on top it. How many times

do you stack of rectangle rectangles to get rectangular prism? It's height amount of rectangles. So that's why it it's actually the base times the height. And this is why I want you to conceptually understand this because the question might not give you the length, the width, and the height. Instead, the question might give you the base of a rectangle and ask you to get the volume. And it gives you the height as well. So it gives you the base and the height. Then you know, oh, it's just base times height. You don't need to know

length time width time height. You know, oh, length time width is is the base. So you can actually like solve questions even though it doesn't give you exactly what the formula tells you. So it's kind of important that you Actually understand where everything comes from. Same thing for the volume of a cylinder. If you think about it, pi squ, right? This area of the of the bottom circle here, right? Pretend like the base of the circle. It's exactly same thing as a rectangular prism. Okay? Like to get a cylinder, you just stack a bunch of

circles on top of each other. It's like this entire area of the circle, you just stack on top of each Other until you you reach height amount of circles stacked on top of each other. And that's why it's it's the base of the circle at the bottom, right? The the area of the circle times the height. This is the circle you stack on top of each other until you get the height. Now, having a conceptual understanding of where these formulas come from is also super helpful for finding the surface area because the SAT loves asking

about the surface area I I think It's cuz it's like a hard thing to do. It's not a formula already in here. So, they they want you to like conceptually understand where formulas come from or at least how surface area works. So, the only shapes I think you need to know how to find surface area of is the rectangular prism, the cylinder as well as the pyramid. And this could be a triangle pyramid or a rectangular pyramid, which I which I'll talk about once we do a practice question with a Pyramid. But for now, let's

go over how to get the surface area of these. So surface area is just the sides of the areas of all the surfaces added together, right? So for a I guess a cube or rectangular prism, there are six sides, right? I'm trying to draw the sides, you know, so the bottom side, this the back side, the top. So there's six sides, right? You imagine a regular prism, a box, there's six sides. You just need to get individually get the Area of each of the sides and add all them together. And be pretty self-explanatory. Now, how

do we get the surface area of a cylinder? Okay, if you think about this, this is why it's helpful to understand like where the volume formula comes from as well. If you think about it, first of all, we can just get the area of the circle up here and the area of the circle at the bottom there, which should be the same. So, we just do pi squ, right? That's going to Give us the the the areas of the of the circles on top and bottom. But how do we get like the side the area

of the side? How we do that? It's actually pretty simple because if you think about it, how we got the volume in the first place was we just took the area. We just stacked a bunch of circles on top of each other height amount of times until we got to the entire cylinder. Right? how we can get the surface area at least like the area of this like round round Part of the side. We just get the circumference which is just a hollow it's just the perimeter of a circle the circumference right it's the perimeter

of a circle. So if we just get the like a hollow circle like nothing's inside like a hollow circle and then we just stack up the hollow like hula hoop I guess on top of each other over and over again that that's going to be a cylinder. It's going to build us a cylinder. just stack a bunch of of us Like hollowed out circles on top of each other h amount of times that will give us this area. So it's really just the circumference times h. Okay, if that didn't make sense to you at all,

you can also think about this as a paper like a piece of paper that's rolled up and then you like unfold the piece of paper, what shape would that turn into? So imagine this is just like a r piece of paper and you unroll it. What's the shape? It should be a rectangle, right? rectangle Is just the length times width to get the area of that. And the length should just be the circumference because if you think about if you roll out the the the paper, you're just rolling out the circumference, right? The the perimeter

of the circle, you're just rolling this part out. So that's just going to end up being the length of this new rolled out rectangle. And obviously the height's going to be the width. So it's just length time length times Width, which is just the circumference times the height. So basically the formula for a surface area of a cylinder is simply just the p square which is just the the area of the circles. There's two of them. So 2 * p squ plus the area of the side part of the cylinder which is just the circumference

times the height. And again I I don't even think you should memorize this. I don't have this formula memorized actually. But I do know how to get the Formula because I conceptually understand how area and volume works and I'm able to just remember this. Like I don't need to think about it. I can just like do it because I already understand how it works. Okay. I hope that makes sense. And you don't need to memorize the surface area for for any other shapes other than I think it's the rectangular cylinder in a in a pyramid.

You need to know how to find that too. It's not really memorization. You just Need to understand what it comes from. But if they ask you for like a surface area of a sphere, they'll give it to you. Okay? That's a crazy shape. I don't even know the area of a sphere. But just make sure you know these three shapes, okay? And we'll go over the pyramid in a bit. It's actually a bit harder than you think it is if you don't have experience with that. But first of all, let's just jump straight into a

question. So see if you can solve this question. Okay, hopefully you pause the video and try solving it. Let me make my head a bit bigger. So a general strategy for area and volume questions is just to get the formula of the area and or the volume or if there's both in this case and get the form of both of them which should be pretty obvious but it it helps you like simplify situation. Okay. So there's always there's always going to be like a area volume because the area volume Equation. So just get the formula

of the area or get the formula of the volume or or both and then just go from it's going to be really easy as long as you do that. Okay. So surface area the cube has a surface area of 54. what is the volume? So, you notice in the reference sheet, it doesn't actually ask tell us that the volume of a cube is is this. It doesn't say that. However, a cube is just a rectangular prism with the length, width, and height being the same Measure. So, it's just length times width times height, but the

length, width, and height are the same number. So, it's just the same number multiplied to itself three times over, which is just that number cubed, right? So, that's that's actually where the the like why it's called cubed. At least I think that's why it's called cube. At least I hope so because like the volume of a cube is just one of the side lengths cubed. Okay, that's the volume Of a cube. And this is also why like you want to understand conceptually where the formulas come from because I've had students actually get stuck on that

because they're like, "Oh, it doesn't tell me what the volume of a cube is. How do I figure that out?" Oh, then you just it's you have to conceptually understand it to figure that out. Okay. And the surface area of a cube. Now, this is slightly harder. So, imagine the cube. There's six sides. All the sides Are the same. You get the area of each of the sides. There's six of them and they're the same measure. So we just get the area of one of the sides multiply by six and you get the surface area.

Right? So the area of one of the sides is let's go figure that thing square is because the the area of a square is just going to be a side length squared and that's why it's called squared. I don't know if that helps you memorize it or not. I don't know. Okay. But that's the Formula. So it's just one area multiply by six because there's six sides. So this area and then we can just solve it now. So like again the strategy was just to get the this one is a pretty easy question but you'll

see a harder question later on but you get the idea just get the volume and the area or the formulas for those and you can go from there and we can just I'm going to use regression turn into a constant but solve uh actually we need to know wait Hold up we need to know what this is first okay and then okay so we know x equals three so length of this cube equals three and now we can plug in three into Q they get the volume 27. So the answer is B for this question.

Okay. Now let's go to a much much harder question. Um good luck on this. See if you can solve it. So pause the video. Try solving this question. Okay. Hopefully you tried at least I figured out a direction to go to solve this Question. Okay. So a figure shown is a right rectangular pyramid. So there's actually two types of pyramids I've seen on the SAT. There's a rectangular pyramid and I've seen triangular pyramid as well. This might say triangular. And basically what that's saying is just the base. Okay, a rectangular pyramid is saying that the

base of this pyramid is a rectangle. That's why it's called rectangular pyramid. If it's a triangular pyramid, Then the base of this this pyramid is just a triangle. Okay. Um there's not really anything special about it. It's just like remember you're solving you're getting the area of this this this triangle instead of a rectangle and you remember the divided by two or something but like hopefully like sometimes it might not give you the picture. So like when you're drawing it out you you know it's just triangle pyramid and it's actually the base of this is

actually a Triangle not a rectangle. Okay. And this right thing, you can just ignore this. This this white right it might say right cylinder, right triangle pyramid, right rectangular pyramid. That's just I think that's just referring to the fact that there's a right angle like it's just like a like a straight up like this is a normal shape, right? It's not like a slanted pyramid or anything. Okay. So it's like if if you if you drew a line so down from the tip to the base, it's It's going to form a right angle. And that's

what it means by right. You can just kind of ignore that because I never test you on things that aren't right pyramids or right cylinders. You can just kind of ignore this. But really rectangular and triangular that kind of matters. So this is the rectangular pyramid. Guys, label the known size. So length was 18. The width is 9. The height is 12. Okay. So got it. So how do we like so I I think we can easily get the area of this rectangle right length times width that's actually pretty we just try to piece together

the formula right so surface area is just all the the areas of the surfaces add together so we can get length times width um which is just what it's uh 18 * 9 right 18 * 9 length times width okay so this we'll get we'll get the area so surface area because we do SA. So surface area equals this. Now we need The the different sides, right? The this side the area of this side I guess the area of this surface. Okay. And there's going to be like this surface is going to be the same

as this surface, right? And like this surface is going to be the same as like the surface that we can't see, right? So, as I guess we could do like sof times I'm going to call this like sofice a like this one the way the one that's facing us the most. So was A and this One. So it's B I guess. So this is 2 * A plus 2 * B and we need to find like A and B. Okay, hopefully I'm not confusing you with all these different things. Hopefully this this makes sense. So

right we just need to get um this is the exact same as the the the base one. So that's why I got so I just need to find I call this a. So we just get the area of this all by two. This is what I mean by like getting the formula. This helps you break it down. So how do we get a? Now I think most of you like to get this formula or sorry the area of this surface a lot of you I'm not sure if most of you did but a lot of

you will just do like height times the the length, right? Because you think oh the area of triangle is is the base times the height divided by two, right? So they they you did like the height like this height and this this length which is the base. That is incorrect. Okay. You would get the wrong answer. It's It's very incorrect. This is a common trap people fall into. If you have a pyramid, okay, that's this this height from the tip of the pyramid to the base like the height that's shown here. The height of the

pyramid is a different number than this the slant height, right? like the like this diagonal thing like the of the triangle like that that of the side we're facing right this this height of the triangle Is a different measure than the height of the entire pyramid it's not on the same dimension okay so you have to actually find it the slant this is called the slant height the the height of the slant of the pyramid okay not the height of the entire pyramid so how do we actually get the slant height then how do we

get this this height. Okay, we can remember there's a right angle here like this. This forms a right angle. It's actually a right triangle right There. That's actually we get this is the hypotenuse of a right triangle. So we know the height, right? This was um 18. We know the height of this pyramid. It's 18. And we can we know this as well because this is just half of the the length which is 18, right? 18. Sorry, wait, this isn't 18. This is 12. My bad. So the height is 12. Yeah, height is height is

12. That that's the value. And then this is going to be 18. So just Half of that which is just like this leg of the right triangle. That's nine. Then this is half of the length. So basically we know two legs of this right triangle 12 and 9. We can get the hypotenuse then using pagorey and the let's just do that right. So we know we need to find the hypotenuse and we know a^2 + b^2 which is just 12^2 + 9^2. Um let's have desmos solve c for us. So c is 15. Okay. So

what do we that's actually This height I think. Yeah. So like the length times the height if like this side thing. Okay. So that's the same height of that. So we know that's nine. Let's just solve for the area of this triangle then because we know the slant height now which was 15 as we found there and we know the width which is 9. So base times the height 9 * 15 / two. So a = 9 * 15 and remember to divide by two because it's a triangle. Let me just time 1/2. Okay. So

that's this is just 9 * 15. So base times the height divid by two. That's going to be the area of this triangle multiply by two because there's two of these like this one and the one this side too. Okay. Now we need to get this triangle which is a different measure than than these right because they they have like a different um base for multiplying like this one base was w now the base is the length and even the slant height is going to be different Now because we don't know. So let's let's calculate

the slant height again. So we use the exact same thing we did before. Let's do that. But the only difference now is it's going to be this. Instead of this length for the leg of the rectangle, it's going to be this length now, which is actually half of the width, not half half of the length. So half of the width, the width is nine. So it's just 4.5 then. So instead of nine, We put 4.5, right? That's going to give us the new slant height because the height's still the same. Like this measure is still

the same. It's just this thing that changed. So we put a new number for that. That's going to be the slant height of this side, right? So, see, it's not a nice number, which kind of scares me, but I'm pretty sure we're doing everything correctly. I'm being really careful with every step we're doing. So, that should be the correct Answer anyways. Okay, so 12. So, C, let's just type in C. Yeah, sure. C times actually, I'm not sure. That's that's fine. Okay, so C equals that value. We just do C times um the C was

the height times the the base which is L. So 18. Okay. And then we times 1/2 because it's a triangle. We multiply by two because there's two of them. Okay. So now we should have the volume sorry the surface area. Okay. The answer is 527.6 Or 7. So, actually, if you didn't know, on the SAT, you can only type in five characters for numerical response questions where there's no multiple choice and you have to type in the number yourself. It only accepts five characters max. You can actually round this number to seven if you want

to. Doesn't matter. You don't have to round it, but it only accepts five characters. So, basically, just it won't even let you type in a six character. By the way, A negative a negative sign does not count as a character. So, if you put a negative sign, I guess you could type in six characters if you want to. The negative, there's no negative. It doesn't matter. Okay? You just type in five characters. Negative sign doesn't count as a character. The decimal point does count as a character. So that's one, two, three, four, five. You can

type in you can round this last one if you want to. It accept both ways. It Doesn't matter. Okay? But it won't even let you type anymore after you type in five. So you just like keep on typing until you can't type anymore and then it'll accept that answer. Okay? So the answer for this one was 527.6 or 7 depending on if you rounded or not. Okay. Now let's do another question. This question should be simpler in terms of like the amount of steps it takes. Okay. But hopefully like you can use the the same

strategy of just let me let me Get like an idea of what the formula is and then just like try to like piece together what the formula is and then you can get the answer. Hopefully we just do this question. It'll make more sense as we do more of these practice questions. Okay. Hopefully pause the video and try solving it. This says a cylindrical can. All right. There's no there's no diagram, so I just have to draw it out. Fine. Okay. So, a cylindrical can Containing pieces of fruit is filled to the top of seal

or syrup before being sealed. Okay. So, basically there's like a cylinder and the only thing inside the cylinder is fruits and syrup. All right. So, there's a cylinder. There's pieces of fruit. I'm just going to put dots as fruit. So, there's a bunch of fruit and there's a bunch of syrup inside this. And there's all the there's just fruit and syrup. The base of the can has an area of 75 Cm squared. So we're given the base. Yeah, this this is a great example of the fact that you can't just like blindly memorize these formulas.

You have to understand where it comes from because it's just the base. It doesn't tell us the radius at all. So if you don't have the radius, like how are you supposed to get this, right? But it tells us the base and we realize oh pi squ is the base. So this is the base times the height. No, it's going to give Us the height, right? Let me check. Um the height of the Yeah, it gives us the height. So the formula for the the volume of this entire cylinder is actually base times height. Okay.

And you wouldn't have realized unless you actually conceptually understood where the formula for this comes from is the base times the height. How many bases stack them stacked on top of each other is height amount of bases. That's like it makes sense to me. Okay? Hopefully it Makes sense to you. So if 110 um centimeters cubed of syrup is needed to fill the can to the top with the volume closest to the total volume. This is closest. That means we can round. Okay, the answer we get doesn't have to be the exact number. If it

doesn't say close or it doesn't like hint that we need to round the answer, then if you don't get the exact answer, you got the question wrong. Okay, you have to get the exact answer for multiple choice Questions unless or you probably made a mistake somewhere. So the following is closest to the total volume pieces. Okay, so let's let's basic strategy, right? We just get the equation of the volume and then we can work from there. So, we're trying to find the volume of the pieces of food, though, but let's first get the volume of

like the the big can, right? We can do that. This is base times height. As we discussed before, it's the base, which is the circle base. How many bases stacked on top of each other? It's it's 10 cm or height amount of bases. So, it's just base times height. Okay, that equals the volume. Now, how do we how do we do this? So, we just need to find the pieces of fruit in the can. And all all that's in the can is just syrup and fruit. And we know the syrup. So if we subtract out

the syrup, what's left is going to be the fruit. Hope that makes sense. Does it make Sense? Because there's only two different things in this this this that makes up the volume of the entire of this entire thing. And if we get rid of all the other stuff, what's left is just going to be the fruit, which which is which is what we're trying to find. And this is actually a really common pattern where you find the area with the volume of the entire thing and you subtract out the unneeded stuff and what's left is

the stuff that you want. For example, um They love asking questions with like shaded regions. For example, let's say there's a big rectangle and then there's um like a shaded region in here. So let's do this. And then there's like a shaded region. Ask you for the area of the shaded region. And how you would solve this question is you just get the the area of the entire rectangle and then you just subtract out the area of the small rectangle inside here and what's left is just going to be A shaded region. Right? So it's

the same strategy where you you you're trying to find something. There's no exact formula to get it. But what you can do is get the area of the entire thing, subtract out the stuff that you don't want, which is just this this mini rectangle here. Then what's left is going to be the thing that you want. So the same idea here. So I want you to recognize that pattern in case you see something like that. That way you don't have to spend Time thinking about how to solve the question. If you see this type of

pattern where it's like, oh, unneeded stuff and what what's left is the stuff that I want, then you could just use use this pattern right away. But this is 640 is the answer. Oh, we didn't even need to like round with anything. It's l was the answer right away. Okay. Yeah, the answer here was 640. Then C. Okay, let's talk about more patterns. So, the next pattern is inscribed shapes. And I want You to pause the video and see if you you would be able to figure out how to solve this question. At least have

an idea of how you could. Okay, hopefully you like read the question and you tried your best to like figure out how to solve this because it's going to make this way more memorable. Okay, there's actually a really good strategy for inscribed shapes. Okay, so in the figure rectangle E FG, so it's just this rectangle, right? Is inscribed in a Circle C. So basically what what inscribed means is just it's as big as it can be inside that other shape, right? It doesn't even have to be a rectangle inside a circle. It could be it

can be the other way around. Let's say there's a square and it's like a circle inside the square. But inscribe shapes is basically meaning they're touching at at like the max like this circle cannot be any bigger inside the shape. Same thing for this. This Rectangle can't be any bigger at least like proportionally. It can't be any bigger, right? And it's like all the edges of the rectangle are touching the circle. That's what inscribed means. It could be like a triangle inside a circle. It could be like any shape inside. It could be 3D like

there's a cube inside of a sphere things like that. Okay. But the general strategy for inscribed shapes is you have to find the common length between the two shapes. So What I mean by that is what length can we work with between the two shapes. And that's if we can if we can find what the common length is. There's going to be one length at least one that's that's going to be in common between the two shapes. And we can use that length to convert between the two shapes. Okay. So in this case which has

a radius of 10. Okay. So we're given we're given hg which is this. And we're given the radius of the circle. Okay. But before We before we deal with that, what is a common length between the circle and this this rectangle? Okay. A good strategy to find this common length is to look at the points where they're touching. Okay. If you notice this point, we just draw a line between these points. You notice that's the diameter of a circle. And it's actually really easy when you're dealing circles to deal with incribe shapes because really all

you know about a Circle is just a radius and a diameter. Like that's the only length we can work with. Radius and diameter. The radius is half the circle. Diameter is the length of the entire circle. Like that's all we can work with. So we just see okay does the radius or the length match something that we can use in the rectangle? And the radius I mean maybe we could use like what what length of this is is the rectangle like how can we use the radius in the rectangle we can't so it must be

A diameter then so if you draw like a diameter just keep on drawing diameters you probably realize oh the diameter of the circle is the same as the diagonal of the rectangle and we can use the diagonal of the rectangle right so we can convert so since we have like 10 radius of 10 for the for the circle that means the diameter is 20 okay so that means the diag diagonal of this rectangle is going to be 20 as well. And that's how we can convert between the Two shapes. So let's say it gave us

information about a circle. We can find the the common length between the circle and the rectangle to convert from length from a circle to a length to the rectangle. In this case, it was the diameter and in the diagonal and we can do okay. So the diagonal is 20 as well. We're also given that this is 16. And what do we need to find? What is the area of E? E F. So like this this triangle we need to find the area of That. Okay. Yeah. So, you notice there's a right triangle there. That's pretty

simple actually. So, like this is 20. That's the hypotenuse of the right triangle. And then we can do here's a triangle. It's a right triangle. What you can do because we're trying to find the area of this anyways. It's the same measure. It doesn't matter. But, um you could you could just think of it like this cuz if this is 16, then this must be 16 cuz how rectangle works, right? It's like these two lengths are the same. So, this is 16, this is 20. Then we just find this missing length. And once we find

that missing length, we can just do base time height. Or I guess you could think of it depends like if you if you orient the right triangle like this is base time height, right? So base time height divided by two and that would be the area. How do we get that? Um that should be 12 I think. But if I if you curious how I did that like how I was Able to like get the side really quickly in my head that's just pagan and triplets because I know three four five triplets. If it doesn't

make sense to you or isn't familiar, just don't worry about it. You can simp which is just um we talk about this is 20, right? So 20 um which is C ^2= A square + B square. Let's say A2 is 16. That's what GH equals, right? It's yeah or HG plus uh FG which is what's called B ^2. And we can solve for B square Which is 12 like I said. So we know the base or you can do is height. It doesn't matter. So, so base time height 12 * 16 / 2. So, 12

* 16 / two. The answer is 96. So, C you would pick and then that should be the answer. Hopefully that makes sense. Inscribe shapes. Just find the common length between the two shapes. Okay? Because to inscribe it, you should it should be pretty easy. Just look at the points that are into That that are touching and then try to like see if you can like draw common lengths. And if you doing a circle, try drawing the radi or the diameter and see if that works and it shows the shape that's inscribed or like it's

it's being inscribed in. Okay, next let's talk about proving triangle questions. And you notice these questions only show up in the hard second module. They don't show up at all in the first module. And I doubt it's going to show up in the Easy second module if you get the easy second module. But because these questions are hard, it's just supposed to be hard. They're actually pretty simple if you know what you're doing. But um like they're kind of like a niche thing that you just have to know what to do. If you don't know

it, then you're gonna get the question wrong. But if you do know it, it's pretty simple. Okay, so the proving triangles, the main thing is proving congruency and proving Similarity. So I I'll go over this. So first of all, what's the difference between congruency? So like proving something two triangles are congruent to each other and two triangles are similar to each other. Now congruent just means they're the exact same shape. Like literally every the sides are the same, the angles are the same. every single thing about the about the the shapes are the same. Okay,

congruent doesn't just apply to triangles, but I've only seen Proving triangle questions on the SAT, but it could be like congruent like rectangles or something. I don't know. I've never seen that though. But you get ideas. Congruent just means they're the same. And then similar, which we talked about earlier, is just one of them's bigger than the other one. Okay, they have the same angle like the proportions are the same, but one of them is just bigger than the other one. That's similarity. Okay, so those are the Differences between congruency and similarities. It's really important

that you know because a lot of people get those confused. Okay. Congruency is they're the same, similar as one of them is bigger than the other one, but everything else is the same. Like the angles are the same and everything. Okay. Now, there's different I guess ways you you can use to prove if something's congruent and different ways you can use to prove if something's Similar. Let's actually go over similar triangles first because we already went over that earlier in this video and we could just do a quick refresher. But, um, there's two ways to

prove similar triangles. There's only two, by the way. Okay. First way is proving that the angles are the same. Okay, all the the corresponding angles are the same. So basically if you have a triangle, you have another triangle and let's say one of the triangles um angles equals 30 60 And 90 and the other triangle also equals 30 60 and 90 like the the angles then we know this those triangles are similar to each other because all the angles are the same. And actually you just need to know two angles have to be the same,

right? Right? Because we know if we know two angles of a triangle, let's say we know one triangle is 30 60 and we don't know what this one is. Like we don't know what the last angle is for this triangle and we know the other one Is 30 and 60 and we don't know what the last one is. This is enough to prove similarity because if you know two angles you can get the third one like 180 - 30 - 60 that's going to equal 90 obviously. So we we can solve for the other ones.

As long as we really as long as two angles are matching two separate angles in in the triangles are matching then we know those two triangles are similar. So that's one thing proving angles are the same. And the other way To prove that triangles are similar is proving that the side lengths of the triangles are proportional to each other. Now this is different from angles. We can't just do two. We have to do all three of them actually. So let's just show you. So let's say we know one of the the sides of a triangle

is two. One of them is three. One of them is I guess see that's not possible. Um I'll show you why later on because you'll learn why that's not possible. I think This is this is probably possible, right? Yeah. Okay. So two, three, and four. Okay. Those are the sides of a triangle. Is that Yeah, I think it's okay. And then that means the and then another one another triangle was four, six, and these are the sides of the triangle again. Okay. So we know one side one triangle has the side lengths 2, three, and

four. And the other one has four, six, and eight. You notice this one like if you look at this this Triangle there's there's always a side length that's two greater than the one before the the smaller triangle right so that means it's always like the scale factor of two it's consistent between all the sides so that means that these triangles are similar now if you only knew one that's not enough that that's not enough you need to know all of them because you really you can have two triangles because you can have a triangle and

like we know these side Lengths but this third limb can be drastically different like it doesn't have to be the exact same triangle. Like it could be this side length or we can take the exact same lengths like this length and this length and make them like for example this and and this and like this third length would be drastically different, right? Like imagine this is the like this is the same as this. So basically I'm just showing you that just knowing two of the Sides of a triangle doesn't automatically determine what the third one

is because we don't know that's only for right triangles where you can use pagan to get the third one. This is just regular triangles. So just knowing two of the of the sides isn't enough. You need all three of the sides to to actually prove that oh all of these are proportional. So it's much easier to prove with angles. But sometimes the question like forces you to prove using The sides and show that they're all proportional too. But that's how you prove similarity. Okay two ways. Okay. Don't get us confused with congruent angles congruent triangles.

My bad. It's only for similar triangles where you can use all the angles are the same or I guess two of the angles are the same and um or all the the sides are proportional to each other. So actually let's solve this question. See if you can do this. This is on the harder side by the way. Just read really carefully and try your best. So pause this video, see if you can do this. In triangles LN and R ST angles L and R each have measured 60. So let me let me draw this out

real quick. L M N L M M N N N N N N N N N N N N N N N N N N N N I'm just drawing regular triangles because it doesn't say the right triangles. Uh this is R. That's a really small R. Oh well, ST. Okay, we know L and R. So I'm I'm not Going to like actually write out this is what I usually like. I just like symbolizing to myself. Okay, these are the same, right? and ln 10 o30 ln and I I know they're not the same length

and this notation means they're the same length but I'm just like doing this just to show okay we know these and by the way I'm thinking this isn't enough because it doesn't tell us any other stuff about the side length we do know that there's like a scale factor of Three here right this is three times greater than this the problem is we need all the side lengths to to be like that like this has to be two times greater than this this has to be two times greater than this so we need like two

more pairs of side lengths but I don't see that I only see like one more pair so what do we eliminate A cuz it's it should be this should be three times greater than this one. This is the same. That's not good. We need to be Consistent, right? Since this is three times greater, we need to have the same proportion between other sides. This one is three times greater, but it's like N st. That's not good enough. We need all three of them. Again, just knowing two of them doesn't automatically mean that the third one

is proportional. So we can limit. So So it has to be C with D. Let's read. The measure of angle M and S are 70. So it says this. No, they have to be the same. These are corresponding Angles, right? If we drew Yeah, like these have to be the same. And it says they're different. So that's fine. Um it might be that like S S is supposed to be this angle. We can work that out. I'll show you what I mean. So like the measure of angles M and T are 70 and 50. So

if we draw this out again because it went away. Um it said M and T, right? So this is L M N and this is R ST. So it told us this was 50, right? And it told us uh m was 70 and we Know these are 60. Okay, so this is actually like a pretty common trick they like to use. This is actually good enough. The reason why is because we know these two angles, we can solve for this one. What is 180 minus 70 - 60? 70 + 60 is 130. And then um

180 minus 130 is 50, right? So that means this angle is 50 and this angle is 50. So that's actually enough. So the fact that it it gave us two separate angles though like it give us this angle and it gives us Another angle that wasn't corresponding to that angle. But that's fine. We can still use the information that it did give us to like prove that oh there's there's at least two angles that are the same for these. Okay, hopefully that that makes sense. And like this one might have worked, but the problem is

like not 180 um 180 minus 70 minus 60 does not equal 60. It equals 50. So like even if S was like this angle at the bottom here and Like it would actually like I messed up the drawing and it should have been like these two angles should have been flipped around or something. It still wouldn't have mattered because that's not 50 or it should be 50. In this case it is 50. By the way, like you don't need to whichever order it it says the triangle in, just draw that in in the same order.

It's like L M N. So like this just means the first the first angle should be L, second angle should Be M, the third angle should be N. And when the first angle should be R, second angle should be S, third angle should be T. As long as you draw it out like that like L M and then you put like exactly so the middle angle like is is at the same spot those are the corresponding angles like the whichever order the angles appears is going to be. So like the M is going to be

corresponding to T. S is corresponding to M. Just the tips for when you draw the triangles. I've Never seen otherwise where it was like oh actually M is corresponding with T. I've never seen this is like the middle angles be corresponding left most angles will be corresponding and the last most angles. So you just draw it out like that. Okay. Now let's talk about uh proving congruency which is um I don't know if it's harder but it's like new stuff you have to learn, right? You've kind of already knew similar triangles. Now, let's go over

congruence triangles. Now, congruency, the only way you can prove congruency is one of these four. And maybe you think, oh, there's four things I have to memorize. Like, what does SAS? This probably should seem familiar to you. It's probably been a while since your geometry days when you were doing this stuff was your like high school. Usually, geometry is like your first or second year, and you're probably like third years. So, I I guess it's not as like familiar, but this Should like seem somewhat familiar because most of you taking geometry by now. Okay. But

really, what does this mean? Like, and how do I memorize this? because there actually a really easy way and a fun way to memorize this. Okay, so basically every if you think about every single combination of SAS S there's like two letters A and S and there's like three three I guess three spots for them. Okay, SAS. So for with three letters and you can only use S and A. How many combinations are there? Okay, there's SAS, ASA, AAS, SSS. There's also a AAA, right? and also um a s as excuse my language but a

ass and I'm not counting about like if you like flip something like sa I'm not counting about that because it's going to be the same like on a triangle um by the way ans stands for side and angle because you forgot so angle angle side like angle and then side right so like If I do if I do like angle angle side that's the same thing as side angle angle it's just like depending like which direction you're looking at it from. So it doesn't it doesn't matter. I I'll show you what I mean later on.

Okay. But just know like this it flipping it doesn't matter. Okay. So the other combinations though a aaa a aaa proves similarity. Okay. It doesn't prove congluency. So that's how so just remember okay aa pro similarity and as Ss or you can think of it as uh sa the all all I remember is that the the SAT is PG so there can't be coord so that makes it really easy because okay there can't be coord is out and similar triangles is out so there's no aa because we're trying to procy so all that's left is

just these four so that's a really easy way to memorize this is like think about all the combinations of a and s for three letters and then just eliminate the Coordinate similar triangles which is a aaa and then what's what's left is just these four that's a really easy way to memorize this okay now what do I mean by like sas so basically it's saying the side the angle right next to that side the the side right next to that angle okay that's this is s and it's basically saying okay if I have two triangles

and I know okay this side is equal to this side and then the angle next to one of The angles next to that side is the same and then one of the sides next to that angle is the same then and these this is the same as this one. Okay, this angle is the same as that angle and this side is the same as that side then that's enough to prove that these triangles are congruent meaning they're the exact same triangle. So if you see this combination like you have two triangles and then um there's

there's sides and then there's angles and there's a side right next to That angle that are the same for the two triangles. they go if you go between them then that's these two triangles are the exact same triangle they're congruent. Okay, same thing. So let me let me just draw the triangles again. I kind of hate how it disappears but I'm dealing with it. So angle side angle angle side angle like this works too. It could also be like angle side angle. It doesn't matter but as long as it's like the this a side right

next to that angle And the angle right next to then that's fine. Okay angle angle s you just like the angle. So this angle the angle right next to that it doesn't matter like this one or this one and the side right next to that angle angle angle S if you if you choose this side this would be angle side angle now that that won't work because angle angle side that would work okay side all the sides are the same okay hopefully that that makes sense let's do a practice question if that Didn't make sense

then I think it would make sense after we do a practice question so try the try this for yourself so pause the video see if you can solve this okay hopefully you try so in triangles ABC let's draw the triangles I I know it can be annoying, but I would highly recommend you draw it out. It's just worth the extra a few seconds to just to guarantee you didn't mess up any angles in your head or anything like that. Just A public service announcement for you. Okay, so here's a triangle A, B, C, and then

D. And by the way, I have seen a question where D was like no additional information necessary was the correct answer. So don't just eliminate that by the way. Um, yeah. Okay. Anyways, so we have this angles B and E. So we know these are the same. Oh, I just like showing that they're the same. Okay, B and E are 27. C and F each are measures. So C and F. So right away we Have an angle and angle. So all that's left, remember the the four angle I'm thinking angle side angle like angle side

angle would work. ASA would work. So here's ASA. Um the other combinations are angle angle side, right? We can do angle angle. If we know like this side or this side, that's enough. Like this side, if it says that this side is the same as this side or that this side is the same as this side or that this side is is the same as this Like any side in general. If it says that, then that's enough to put congruency. Um angle angle angle won't work because that's that's similarity, right? So like this angle this

wouldn't work because that's similarity. I think that's all the combinations out of the four because the other ones are side side that won't work. So we have two angles, right? And the other one is side angle side. We have two angles already. We don't then then we would need like Two. We need like this side and this side or like this side and this side. That's so we need like a two sides and an angle in between though. And I don't think that's going to work, right? It just gave us two angles. We just probably

just need one sides. If you look at the answer choices, yeah, angle doesn't work. That's only one. There's not even like two. It's not like this angle. This angle is the same. It's not even that side. Look, that's not even Like two sides. Oh, they're making it super easy. Okay. The length of sides BC. So BC. Yeah. And yes, that's angle side angle, right? angle side angle. There's two angles. We know these angles are the same. This side is the same and this angle is the same. Bam. That's angle side angle. So, we have a

way to prove that they're congruent angle side angle because it's one of the four that proves congruency. So, um C would work. That would prove that these two Triangles are congruent. Okay. And finally, so we we went over proving similarity, proving congruency, but now let's go over how to like actually prove it's a triangle because like there's actually like rules for there can only be this many this this length of sides for a triangle if that makes sense. Like the third side of this triangle can't be greater or or less than this length. And this

this is called the triangle inequality theorem. Now, you actually don't need to know this theorem because for every single at least digital SAT question I've seen. I I remember back in the paper SAT, I think you had to know this, but I'm pretty sure you don't need to know this now. Like, you generally don't, but you do need to know the trick that they they test you on involving this thing. So, here's like a good question to simplify that. So, I just want you to try this because it's better if you like make the Mistake

and then and then you'll learn from your mistakes. So, I want you to try this and like get the question wrong and then that way I can tell you, oh, this is actually how you do it. It's going to stick in your brain much longer. So, um, now that I said that though, you probably won't make the mistake. See if you can like beat me and like not make the mistake. So, pause the video, see if you can solve this. Okay, let me solve this question now. The triangle inequality theorem states that the sum of

any two sides of a triangle must be greater than the length of a third side. So, that's just the triangle inequality. I'm going to explain this as as we read on. So if if a triangle has side lengths of seven and 10, which inequality represents the possible names of x of the of the third side of the triangle. So if we have a triangle and we are told that this is seven and this is 10. What can be this This length? It's not like this length could be like a thousand, right? Like that's not possible.

Like if you think about like even if I stretch these two sides to the ve very minim the very maximum like I just like stretch it's pretty much a straight line at this point. I can't connect like this a thousand length side. It's gonna go off the screen. I can't connect the end points of of the the end points of this this line, I guess, to the the rest of The triangle. It's not possible. So, there has to be a limit to what the third side can be. And if most of you at least watching

this video probably thought like seven, it has to be a number less than 17 because 7 + 10 like two any two sides of the triangle, right? So, seven must be greater than the length of those sides. So basically saying if I add up two sides of the two sides of the triangle. So 7 + 10 that must be greater than the Third lane. So the this third line can't be 17 or or greater has to be a number less than 17. And then you picked a be number less than 17. However, that's actually not

right. But you are right on the fact that this side length has to be a number less than 17. However, what if the sign was one? Well, that's still would that still satisfy a triangle in quality film because remember it says any two sides of a triangle. It doesn't say that it has to be specifically these Two sides like seven and 10. It could be that this side length and this side like seven and 10. It could it could be that it's this side this side. If you add these two side lengths seven plus one

that equals eight which is less than 10. So it can't be one actually. So there actually is a a minimum too. Not just a maximum but a minimum side length for this triangle. Okay. So what is that minimum? I think you probably already like got the answer By now. But um so if I like what I had to figure out so like 10 10 is obviously the maximum. So if this isn't the maximum side length and that that must mean 10 is the maximum like the the largest side length on this triangle. So that must

mean that this side length has to be um like seven plus this side length has to be greater than 10. Okay. So 7 + 3 would equal 10. So that means this third side length has to be a number greater than three as well. Okay. 7 7 and because it has to be these two side length sum has to be greater than 10. So we can't do this. So it has to be like some number greater than maybe four like four plus 7. That's fine. That's greater than 10. So the correct answer to this question

is actually C. And congrats for the people that did figure this out and got it correct the first time because this is a really hard question. It's pretty much a trick Question. It's really hard to recognize that because most people just assume it's a immediately and um maybe they forget to check the other answer choices or they forget to consider, oh, this is might be the lowest thing. Well, now you know, now that you've experienced this trick, you won't fall for it during the actual test. So, if you see something to do with triangle inequality

like these two side lengths must be greater than the third length like that, then you Know, okay, is this also the the greatest length also the smallest length? It could be account for the minimum and the maximum side that this third length could be following this field. Last and definitely the least is circle characteristics questions. They used like pretty much never appear at all for some reason. They used to appear a ton on the paper SAT. I don't know why on this SAT they pretty much never appear Now. But I they all there are a

lot of circle characteristics question at least what I like to call characteristic questions on the question bank by college board. So it's not like they're not going to test this. They could definitely test you on this. So I would know this. It's just they don't show up on the actual test for some reason like at all. But um I I think it's helpful to learn there's only been one sort of characteristic question at least what I'm about to go over one of these questions in like the past 12 13 I think around official tests and

that's why there's only in the first module there's like a pretty easy question actually it been like a really hard one they used to be all over the place in the paper SAP. Yeah. Okay. I don't know but I'll just go over this just just in case it does show up and you know this. So by the way when I say circle characteristics I'm not talking about circle equations. Circle equations actually are pretty common. They show up all the time. That might be like like what they listening to is they want to do more circle

equation questions for some reason. But I'm talking about like normal circle questions that you can't solve with Desmos. There's no like equation of a circle. There's no unit circle thing going on. You can solve those with Desmos which I have like the Desmos guy. I have a separate video just for unit Circles that you can watch on my YouTube channel if you haven't done that already. I'm talking about circleistics just for like the questions like the the normal non-desmos questions that you can't really solve with desmos from start to end. you actually have to know

like a bit of geometry to solve the question. >> Okay. >> Um for these questions I break it down into I guess like two sections. There's Arc length to sector area as well as just like random characteristics you have to know. Okay. So it's just I guess two different areas. So let's go over arc line sector area first which uh I think a lot of people get this over complicated which is actually pretty simple if you just know this formula which is basically the central angle of a circle which so that's that's just like the

middle angle right central angle is just the the angle at the Center the central angle the central angle so the yeah the angle at the center right it's pretty self-explanatory the central angle over the the angle 360° Right? So the entire angles if you don't know it's also in the reference sheet really tells you here right the the number of degrees of arc in a circle is 360 the number of degrees in radians is 2 pi but basically the entire angle of a circle adds up to 360°. Okay so that's that's where 360 comes from.

So the central angle over the entire angle of the circle is equal to the arc length over circumference. So it's it's equal to so this over like all this entire angle is equal to this over this entire um circum I know it's not like this entire circumference of the circle and it's also equal to the sector area. So like this area thing over this entire area. Okay. So if if you can't tell already like these sector area Questions have to do with like circles looking like pizza pies. So if you see like a pizza slice

type of thing, this is a you have to use this formula pretty much. And I think this you don't even need to memorize this formula because I think this makes logical sense. But like if I if I have it makes sense that the fraction of this over this is the exact same as this over like the circumference of the circle. It's exact same as the area over the entire area of the circle. It just makes it looks like they're proportional to each other, right? Like you just imagine like it's it should be like I think

it it should just click that. Okay, that makes a lot of sense, right? Okay. Um, you can memorize it if you want to. I don't think it's like I just want you to understand this to make it make sense to you. Yeah. It just like looks proportion like draw them out like just look at them. You should be able to tell that makes sense. It's the angles On top of entire circle angle the length of the circle on top of the entire length of the circle and the area of the circle of a portion of

the circle over the entire area of the circle. Like all of those are the same if they're like within the same I guess pizza slice, right? Like this angle, this arc and the area of this arc. like it's it it's the the same right for like pizza slice proportionally to the entire circle. Okay. Um I'm like going deep because I Like want you to understand so that way you don't have like waste of brain power to actually memorize it. I think it's much easier if you understand what's going on. But it's actually like a really

good strategy for questions like this. If you see a question having to do with like a pizza slice or anything, all you have to do is just like mark all the stuff that you do know. Okay? Like the the question tells you. So it often tells you oh what s angle if you know The central angle you already know it's a 360 by default because that's just a number and let's say it gave you the radius and you can use the radius obviously to get the circumference because that's like in the formula for circumference and

circle area too. So all you need to find the area of a circle and circumference of the circle it's just the radius. So let's say the question gave you the radius. So you automatically knew the circumference and The circle area and the question gave you obviously know 360 and the question gave you the central angle. Then you can you can just solve for this now. So I can just like put this in Desmos. I can also like solve for this now because I can just put this equal to this on Desmos. It's going to solve

for that for me. So basically all you need to know is like any of these two fractions. So maybe it's these two, maybe it's these two, maybe it's these two. And you need To know three of the four measures of these. And like most commonly it's going to give you the radius. So just by if it tells you the radius, you automatically know the circle area. So really you just need to like know one of these. Okay? I might not always give you radius by the way, but the just a general strategy wise. So I

just like have this formula in your head while you're reading the question. Just like mark off, okay, I know this. I know this. Okay, I I can Solve for this then. Then you can just sol because you already know this by default, right? You can just solve for that. Um I show you examples though. So let me let me put this formula like right underneath the question so you can see it while you're doing this. Okay. So let me see. So pause the video once I like show the question. in its full glory, I guess.

So, pause the video, see if you can solve this question. Use this formula. Um, better Yet, try to like memorize the formula and like I guess like use your hand to block the bottom of the screen or something. I don't know, but try to like it's better if you can just like already have the for you understand where the formula comes from so you already know it. But anyways, pause the video, try solving it, okay, hopefully you had a chance to look at it. Um, let's actually Yeah, let me just keep this here. Yeah, why

not? So circle above has a center O The length of arc A D C. So A D C that's the length. So is five. So it really told us the arc length. We already know that X. So that's the center angle. So we already know 100. Okay. What is the length of arc A B C this arc? Okay. Okay. Interesting. So we we know this obviously. Let me zoom in so you can see. So in this formula let me let me go to the one top. So in this formula we obviously know 360. It told

us that this was 100°. It told us That this was 5 pi I think maybe 6 pi I don't know but it told us this length and we need to know like the arc length that it's that was like the other way around like this one. How can we get that? Um if you just look at this like this the stuff we know what what can we find using the stuff we know? We know these three like what else in this formula can we solve for? Obviously the circumference, right? Because we like these two fractions,

we know three of The four values. We can obviously just like put this as x or something on Desmos and just have Desmos solve this for us. We can easily get a circumference. And once we get a circumference, that circumference is just the perimeter of the circle, right? It's just this this entire circle. Then we can just subtract the perime like subtract this part, which we already know is 5 pi. I subtract this part from the circumference. Remember like the Strategy of the area and volume where you subtract the stuff that you don't want to

get the stuff that you do want. you subtract it from this and then you can get what's left. That's easy. And then you can just get the answer, right? So this is where the formula comes in. So this is like a pretty standard um process. You just follow the formula, right? So let's type that in on Desmos. Who needs mental math, right? So um central angle was Um my des central angle was 100 over 316 because say like at least this this fraction or this proportion should be equal to the same fraction or the proportion

of 5 pi over the circumference output c whatever for circumference and then we can see what the circumference is and then we can literally just do now that we know the circumference the entire perimeter of the circle we can just do um c minus 5 pi and then this is Going to be what's left the answer. Now, I'm afraid that yeah, these are in pi. So, I guess we'll type in the answer choices until we see which one. I think it's probably like uh a 9 pi. Shoot, I'm stupid. 13 pi. Yeah. Okay, that was

close enough. Okay, so the answer is B for this one because like 13 pi answer is B is the same as the arc ABC, which is what we were trying to find. Now, that's pretty much it. Okay, like hopefully this seems pretty simple to You now. This arling area questions and they're pretty uncommon that you'll see this but now you know in case you do see it. And just one more thing is actually like super simple. If you see something to do with arc angle okay there's a difference there's arc length this arc length like

the the actual measure of the length of this arc also the arc angle which is different from the central angle. Okay, central angle is like the angle on the center of the Circle. But if it tells you if it meant you something about the arc angle, it's actually really easy. Arc angle is just equal to the sensor angle. So if it tells you tells you that this arc the angle of this arc is 100° or something, then all you know, okay, this is 100° is wrong. Like I told you this and ask you like what's

the arc angle? Then you oh it's 100°. So the central angle is 100°. Basically the arc angle is equal to the central angle. Is I guess it's like Another way to say the central angle or something. But I've literally seen questions tell you, oh this is the central angle. What's the arc angle? And you have to like tell Desmos or tell tell the SAT what the arc angle is and it told you like these are going to be a really easy question but a lot of people just don't know that the arc angle is the

same as central angle just in case you know and it's always like the one like you can't do like this arc angle is The same as this one obviously has to be the one like in that same pizza slice obviously okay that's ar angle and that's sensful angle and arclename sector area questions let's deal with more types of circle questions This is even less common that you'll see questions like this, but we'll go over it just in case you need to know them. So, if if there's a circle on your like the question has something

to do with a circle and it's not there's no pizza Size or there's nothing or like you don't it's not a name sector area question or like you need more stuff to like be able to solve the question basically like you need more information somehow even though because the question doesn't give you enough. It's I guarantee you you it's one of these four. This is just like with right triangles where like you have a checklist of three things also profag and field special right triangles let's See one of them applies this is just like this

a checklist of these 14 so you have to memorize this actually you don't memorize the first one technically because this is in the reference sheet which is just the sum of the central angle of a circle measures um 360° or 2 pi radians which is lally right here right it says the degrees of arc in a circle 360 or 2 pi radians so it already tells you that but I guess you only memorize the first one. I think it's Pretty obvious. Pretty much everyone knows this, but you definitely need to memorize these three. At least

I would memorize these four as like a checklist you need to go through. If you see a circle and it's like a you don't know the answer right away, just like go through these four and just see which one applies, which one could apply and then just go through that. Okay. The second characteristic of a circle that you need to know is lines tangent to a Circle and the radius of the circle form a right angle. So what I mean by that is if I have a circle and there's a line by the way a

tangent line just means it brushes it one time like there's a line barely brush it like this this isn't tangent this is because it brush it two times but this is a circle and like a line barely grazes the circle that means it's tangent to it so you should it should say the word tangent by the way in the question if you see a circle and You see the word tangent then that's a dead giveaway so let's say this is tangent like pretend it only touches at one point on the circle that's tangent so what

I'm saying is this line the line that's tangent to the circle and the radius radius of the circle form a right angle and this is helpful because like it's a right angle you can probably make a right triangle somehow or like use that it's 90° so you can use that angle to solve for other angles things like That so again if you see lines tangent to a circle automatically you should be thinking oh uh like I should probably draw like a a radi and then that was a right angle and use that okay third one

is I think you like intuitively know this but it's less obvious the radi of the circle had the same length Okay, like all the radiuses or radi plural have all the same length. So basically any triangle that contains two radi is an isosesles triangle. So if I have a Circle and I have a triangle inside the circle that's made up of two of the radi. So let's say this triangle, this is a radius and this is a radius of the of the circle. This is a isles triangle. A lot of people don't realize that. So

like a good strategy when dealing with circles is just to draw a bunch of radi in the in the circle. Just see if you can form a sauce triangle. you work with that cuz with aoses triangles again all you need is just one angle on that Triangle and you can solve for the other angles that might be helpful. trying to find the angle of something in the triangle and you know you know for example um like this angle and but you have no idea okay how do I get the other two you just need to

realize oh this isles triangle because the these are these are radi are the same length right so this isles triangle like oh I can solve for these two because isoclesles triangle I can just I can and we went Over how to how to get the other two angles but they have the 180 and like this is the same as that one I can just do 180 minus those two to get that I get hopefully that makes sense. Okay. And the fourth one is called inscribed angles. When the actually I have a diagram of this let

me let me scroll down. So inscribed angles is basically when there's the central angle and an angle on the circle and these two angles share an end point of the angle like They like the I guess the ends of the angle if that makes sense like the ends of the angles they show an end point. So again if there's a central angle so an angle on the center of the circle and an angle somewhere on the circle like anywhere on the circle and they share end points okay they share like end points. This this means

that this angle will always be exactly half of this angle every time. This is called inscribed angles. By the way it doesn't Have to like look like this. By the way this this is if this is like 40° then we automatically know this is 80° right? cuz this is always going to be half of this like the the angle on the circle is always going to be half of central angle if these two angles show end points. Okay? And it doesn't look like this. It can even look like um this is the central angle and

then this this is the it look like this as well. It's the same thing, right? This is on the circle. It Shows the end points of the central angle. Then that's fine. This is going to be half of this angle. It's always going to be the case. Okay, let's do some practice questions. I think this question is too easy. Let's do like hard ones. Let's do some practice questions. You know, these questions can get pretty hard. So, I guess count yourself lucky that you don't see on the on the actual official test that often because

they can get pretty hard. Okay, so pause the Video, try solving this question and see if you can solve it. Okay, hopefully you tried solving it. It says in the figure above point O is the center. So, we know that's central angle basically um of the line of the circle line segments LM L and the tangent. So, right away you see tangent immediately all right angles. Yeah, these these are 90° right? I see these lines are tangent to the circle um and respectively and the segments Intersect a point M as shown like there at 60°.

If the circumference of the circle is 96, what is the length? Okay, so I'm I'm actually thinking of arc length sector area questions as well. The circumference and and everything but we know these are 90° because of tangent. Then that means we can solve for this, right? Because one way you can go about this is like divide this in half. Then it's 309 triangle. This is 90°. is you divide the 60 in half that's 30° 30 60 90 that means that this must be 60° so this entire must be this entire angle must be 120

or you can just do like oh quadrilateral four sides I know like if a if a shape has four sides and all the angles add up to 360 you could have done that as well and 360 - 90 - 90 - 60 equals 120 basically we can get the central angle okay and like remember our formula for the for this right if we got the central angle and it told was the circumference. I literally Said what the circumference was. We know such angle and obviously we know that the angles of 360. Then we can just

solve for the arc length. Were we trying to find the arc length? I don't even know if I read the entire question. What are we trying to find? Um what is the length of minor arc ln? Yeah. Yeah. We're trying to find L. By the way, it says minor arc. It's just the small because there technically two arcs like two there's two lns, right? This this ln Is also like this ln. So the minor one is just the one that's minor. the one that's less the major one. If it's a major arc, then it's going

to be the one that's bigger. This is minus. So, it's this one. Okay. So, let's just solve for it because we already know like the angle and we know um the circumference which is 96. Let's just solve for it. Um so, central angle which was 120. We found that over 360. I'm just using the formula we talked about. Okay. Equals We need to find this right and 96 32. So, 32 is the answer. That's how you solve that question. I know that might the hard part about this is like recognizing oh tangent line forms a

90 degree angle but if you know these points you do a checklist okay which one applies oh it see tangent so we can probably use this then the question pretty much solves itself after that you just have to like mess around a bit like think about it a bit okay but like I Would try to work backwards maybe how do I get this or I need this angle how do I get that angle or I need to like get from these angles to get that angle that's pretty easy and then you can go from there

working backwards helps a lot okay this is the last question we're going to do this is pretty hard question like I genuinely this this is one of the hardest questions on a test at least before like I knew what I knew before like I remember trying to solve this Question and like taking hours for me to figure it out. So like don't be um ashamed if you can't figure out what the answer to this question is but try it. So pause the video give yourself a challenge if you can solve this and then we're going

to go over it. Okay hopefully you at least attempted to like think through this. It says point P is the center of the circle in the figure above. What is the value of x? We just need to find this. Okay, this we just Need to find x and like we're given all this. So um right away you probably thought inscribed angles but the problem is like how do you get any of these? Incredible angle would just mean okay this is going to be double of this angle. The problem is how do you get this angle?

We know these are 20 but like how does that translate to this? You might think oh this looks like it could be 20 as well and you put 20. I don't like you Have to actually prove it. You have to prove like what this angle is. You can't just do it and you probably like probably So remember like my tip for let's let's go for the checklist actually before I like spoil too much to help you connect dots. So first thing someone central angle measures add up to 360 or 2 pi in radians but um

I don't think that helps you much at least right now cuz like there's really no angles around here to add up to 360 anyways. That doesn't help you. There's no tangent lines so the second one doesn't matter. Now the third one all rad of the circle have the same length. Any triangle that contains two radi is an isoclesles triangle. So this is a we can try using this right we it doesn't hurt. So can we form isoclesles triangles with radi. So let's try drawing like a different radi. Um so we have radi here radi here.

We can try drawing this that forms isoclesles Triangle. Does that help us at all? We know it's 20 and that doesn't mean like we can get this angle. I don't think so. Yeah like there's no way we can get that angle. It doesn't help us at all. But let's try drawing other other radi. Okay. Another isosles triangles I I think we can make is this. We draw a line from here to here that forms that makes this triangle as well as this isoclesles triangle. And now we can use this right because triangle we know one

Of the angles in this isoclesles triangle. So we can get the other two. So we know oh this is the same as that one and these are both 20. These are both 20. Oh that means this is 40 because 20 plus 20 is 40. And then you can use inscribed angles right remember inscribed angles when the central angle and um basically these this central angle and the angle on the circle still end points right then the central angle is going to be always Twice as big as the angle on the circle or this angle is

going to be half of this this one however way you look at but we know this is 40 that means that this must be 80 then and this this is the answer if for some reason you didn't know inscribe the angles is this like a quicker way to solve the questions like this. If you if for some reason you didn't know incred you could still solve the questions. We know this is 20. That means that this must be 140, right? Cuz Uh 20 minus or 180 minus 20us 20= 140. This is like finding this angle

from this angle because it's triangle. We can get 140. We know this is 140. And then we can use the sum of the angles at the 360 rule to do 360 minus 140 minus 140 which is going to be 80. That works too. Okay, you didn't have to know gracing was like a faster way you can solve questions like this. Okay, that's it. That's pretty much all for geometry and trig on the SAT. And congrats if you Made it to the end of this video. You you technically learned how to solve 15.55% of the questions

in the first module and 12.82% of the questions in the second module. And pretty much every single geometry or trigonometry question that can't be solved with Desmos from start to end, you know how to solve now. And you can watch my Desmos and my unicircle video if you want to like learn the like the other questions that that like Technically all geometry questions technically but can be solved with Desmos. You can watch that. But this is this is pretty much it. You you know how to solve every single question. Now it's just like using the

tools because there's a difference between knowing how to do something and actually be able to do it. Now you just need to practice using these in in real time and like being able to recognize oh this is how I solve the question faster. Well, I hope They gave you a nice framework of like how to think about how to solve questions like what to look for. For example, for right triangles, if I see a square root of three, I automatically know it's going to be 30 60 90 triangle. Um, for area and volume, oh, let

me just try to get the equation of this this area first. Okay? And then you can work your way backwards like figure out what you need to get and and then work backwards like triangles or these Triangles. I just need to find one angle on here. Hopefully that makes sense. I give you different strategies you can use and different checklistes to to run when you see different patterns. And um good luck on the SAT.