Pythagorean Theorem & Basics of Triangles - Video Tutorials & Practice Problems

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Review of Triangles

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Everyone. Welcome back. So we're going to spend a lot of time in this course talking about angles and triangles. And I want to give you a really good solid foundation for this because we'll be talking about them a lot later on. So in this video, I'm just gonna walk you through a basic sort of refresher on the basics of triangles. There's a couple of important conceptual and mathematical things you need to know some of them will do some examples together. Let's get started here. A triangle is really just a geometric shape with three sides. That's a really sort of basic definition over here. All these things here have three sides and they all sort of close together to form a shape and that's a triangle. Now, uh there's actually three types of triangles that we can classify based on the lengths of their sides. This is, this is just some vocab that you'll need to know. All right, the first one is called an equilateral triangle. And this is where if you'll see all of the lengths of the triangle are the same, all of them have a length of three equilateral, just means that three sides have equal length. That's actually what that word means. All right. The way that we indicate this in diagrams is you'll see these little tick marks next to the numbers that just means that those two or three measurements are all the same. That's an equilateral and isoli triangles. The next type. And this is where actually two of the side lengths have equal length. Notice how the bottom side is three and these two top sides over here are five. That's an isosceles triangle. The last one is called a scaling triangle. This is where actually none of the sides have equal length. Notice how this is a three, this is a five and this is a six. So you'll see no tick marks anywhere. Another way of saying this is that all the sides are different in the scaling tribe. All right, that's really all you need to know about their sides. Now, whenever sides meet in a triangle, they actually form angles. So the way we indicate this is by using a little curved arc symbol over here, and we express that angle in terms of degrees. So whenever you have two sides of a triangle meets, they form angles and there's three other types of triangles that we can classify based on those angles. All right. So again, this is just more vocabulary over here. An acute triangle over here is one in which all of the angles are less than 90 degrees. All of these words that you'll see acute obtuse and right have to do with what those angles are relative to 90. So look at this angle, the these uh these angles over here, all of these things are less than 90 degrees. So this is an acute triangle, right? So this next one over here, you'll see that there's 22 angles that measure 35 degrees, but there's one of them that measures 100 and 10. And this is an example of an obtuse triangle because one angle is greater than 90 degrees. So that's an obtuse triangle. The last one is called a right triangle. We're gonna be spending a lot of time talking about these and these are special triangles where one of the angles is exactly equal to 90 degrees. All right. Now, regardless of any type of triangle, whether we're looking at the sides or the angles. One really important thing you need to know is that in any type of triangle, all angles will always add up to 180 degrees. That is a fundamental property of triangles. So you would look at all of the triangles over here. All of these three numbers will add up to 100 and 80 same thing for this and same thing for this. All right. So that's uh the really important because if you know that all of the angles add up to 100 and 80 if you're ever missing one of them, then you can always find the other one. All right. So that's actually really important. Let's go ahead and take a look at our first example over here, we're gonna, we're gonna um for each of these triangles, we're gonna figure out the missing angle or the missing side. All right. So in this case over here, for this first example, we have a missing side represented by a variable over here. This is X. How do we find that? Well, we haven't discussed any mathematical ways of calculating this. But one of the things you could notice here is that these tick marks uh mean that the measurements have to be the same. So in other words, if the left side is four, then that means that this also has to equal four. All right, that's just something that you might need to know. Let's take a look at the next one example. B this is one where we have two of the angles, this is 40 degrees and 40 degrees, but we're actually missing one of the other ones. How can I find that? Well, again, remember all of the angles have to add up to 100 and 80 degrees in any triangle. So if you're ever missing one of them, you can always find for the other, uh I'm just gonna set up a simple equation over here. This is gonna be 40 plus 40 plus theta. So in other words, if I add up all of the angles I have to get 100 and 80. If I just subtract 40 from both of the sides over here, this is basically the same thing as subtracting 80. What we're gonna find here is this angle is equal to 100 degrees. All right. So this angle over here is 100 degrees. And therefore, this would actually be an obtuse triangle, but that's not what the question asked us. All right. But that's the answer, theta equals 100. So that's it. That's just a basic introduction. Let's go ahead and get some, get some practice.

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Problem

Problem

Classify the triangle below according to its sides and angles.

I. Equilateral

II. Isosceles

III. Scalene

IV. Acute

V. Obtuse

VI. Right

A

I and IV

B

I and V

C

II and V

D

II and IV

E

III and VI

F

Only one of I, II, III, IV, V, and V

3

Problem

Problem

Find the missing angle $\theta$ for this right triangle.

A

30°

B

60°

C

90°

D

120°

4

concept

Solving Right Triangles with the Pythagorean Theorem

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5m

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Everyone. So in a previous video, we discussed the basics of triangles and I mentioned that we would very commonly be working with right triangles. But one of the most common situations that you'll see is where you have two sides of a right triangle that are known, but you have an unknown side. So for example, we have three and four that are known here, but this side X here is missing. Well, don't worry because in these kind of situations we can always solve for this missing side by using something called the Pythagorean theorem. It's probably something that you've heard before in a math class, but we're gonna be using it a lot in this course and you'll need to know it. So I'm gonna go ahead and explain it to you. What I'm gonna show you. It is that it's really just an equation relating the three sides of a right triangle. Let's go ahead and get started. We'll do some examples together, all right. The first thing you need to know about the Pythagorean theorem is you can only use it when you have a right triangle. All right. So you can only use it when you can assume or when you know that one of the angles over here is 92. If you don't know that, then this equation won't work. So what is the equation? Well, it's really just A squared plus B squared equals C squared again. You've probably heard that before. But what it just really means is if I take these two numbers over here A and B, they'll just be numbers and I square them and add them together. That's the same exact value as this side over here, squared as well. Let's take a look at our first example. So we can actually just get some practice with us and do it together. All right. So we have three and four that are known over here and you have X that's unknown. This is the right triangle. So I'll be able to use the Pythagorean theorem to solve that missing side. I just have my equation over here. A squared plus B squared equals C squared. All right. So how does this work? Well, what's really, really important about the Pythagorean theorem as well is that you always have to take mind, uh take notes that you're A and B need to be the shorter legs of the triangle. Always set your A and B as the shorter legs that form the corner, that form that 90 degree angle. And then you wanna set C as the hypotenuse the hypotenuse of a triangle is always the longest side, which usually is gonna be the diagonal. Not always, but it's almost, almost gonna be the diagonal one. All right. So in this right triangle over here, what we can see is that these two form the sort of corner like this, that's A and B and C is gonna be the diagonal, the longer one. So that's what we set as C and when it comes to A and B, it actually doesn't matter which one you pick as A or B, I'm gonna go ahead and just pick this one as my A and this one as my B. So what this equation says is that A squared plus B squared is C squared. So in other words, if I take four and I square it and I add it to three and I square it and I figure that out, that's gonna give me this missing side squared. That's gonna be X squared. All right. So four squared plus three squared. This actually just ends up being 16 plus nine, that's gonna equal uh X squared. And if you actually just go ahead and work that out, that's gonna be 25. So are we done here is the answer just 25? Well, no, a lot of students will mess this up. You have one last step to do here, which is, you have to take the square roots of both sides because you want X not X squared. If you do that, what you're gonna get is that X is equal to five and that is the answer. So the hypotenuse of this triangle is equal to five. All right. One really quick way to sort of check your work is that number of hypo news has to be the longest side. So notice how five is longer than three and four. If I got something that was lower than three or four, then I know I, I would have messed something up. All right. But that's really all it is. All right. That's the Pythagorean theorem. Go ahead and pause the video and see if you can find the answer to this problem over here. Example B All right. So let's go ahead and work it out. So we've got this triangle over here. Notice how it's actually a little bit different because we have a right triangle just like we did over here. But this time, we actually know what the hypotenuse is that diagonal, longest side is something we already know. And in fact, one of the shorter legs is actually one of our unknown values. But we can, we can still use the Pythagorean theorem because remember it's just we know two sides out of three. I'm gonna start off with my equation over here. A square plus B squared to C squared. I set my C to be the hypotenuse. So the word C is 10. And then again, it doesn't matter which one is my uh missing variable right or sorry. Which one is my A and B, either my A and B will just make up that 90 degree angle. So I'm gonna just go ahead and set this one to be A and this one to be B, we'd get the exact same answer if you did it the other way. All right. So this is A B equals six and A equals Y. So what the Pythagorean theorem says is that A squared plus B squared, some of the words Y squared plus my B which is six squared, that's gonna equal C squared and C squared is just 10 squared. That's gonna be 10 squared over here. Now, you just plug in the values that you already know and just sort of calculate right. So this is gonna be Y squared. Uh This is gonna be plus 36 that's 36 over here. That's gonna equal 10 squared, which is 100 you can subtract 36 from both sides like this, subtract 36. What you're gonna get over here is that Y squared is equal to 100 minus 36 which equals 64 that's equals 64. Now, the last thing you're gonna have to do, let me actually just write that in blue. Last thing you have to do is you just have to take the square root of both sides. You have Y is equal to the square root of 64 and that's equal to eight All right. So that is equal to eight over here. So that means we go back into our diagram and this is gonna equal eight. Notice how again the hypotenuse the 10 is still longer than both of the other, shorter sides over here. And that perfectly makes sense. All right. So that's it for this one, folks. Thanks for watching and I'll see you in the next one.