High school geometry
Showing the ratios of the sides of a 45-45-90 triangle are 1:1:sqrt(2). Created by Sal Khan.
In the last video, we showed that the ratios of the sides of a 30-60-90 triangle are-- if we assume the longest side is x, if the hypotenuse is x. Then the shortest side is x/2 and the side in between, the side that's opposite the 60 degree side, is square root of 3x/2. Or another way to think about it is if the shortest side is 1-- Now I'll do the shortest side, then the medium size, then the longest side. So if the side opposite the 30 degree side is 1, then the side opposite the 60 degree side is square root of 3 times that. So it's going to be square root of 3. And then the hypotenuse is going to be twice that. In the last video, we started with x and we said that the 30 degree side is x/2. But if the 30 degree side is 1, then this is going to be twice that. So it's going to be 2. This right here is the side opposite the 30 degree side, opposite the 60 degree side, and then the hypotenuse opposite the 90 degree side. And so, in general, if you see any triangle that has those ratios, you say hey, that's a 30-60-90 triangle. Or if you see a triangle that you know is a 30-60-90 triangle, you could say, hey, I know how to figure out one of the sides based on this ratio right over here. Just an example, if you see a triangle that looks like this, where the sides are 2, 2 square root of 3, and 4. Once again, the ratio of 2 to 2 square root of 3 is 1 to square root of 3. The ratio of 2 to 4 is the same thing as 1 to 2. This right here must be a 30-60-90 triangle. What I want to introduce you to in this video is another important type of triangle that shows up a lot in geometry and a lot in trigonometry. And this is a 45-45-90 triangle. Or another way to think about is if I have a right triangle that is also isosceles. You obviously can't have a right triangle that is equilateral, because an equilateral triangle has all of their angles have to be 60 degrees. But you can have a right angle, you can have a right triangle, that is isosceles. And isosceles-- let me write this-- this is a right isosceles triangle. And if it's isosceles, that means two of the sides are equal. So these are the two sides that are equal. And then if the two sides are equal, we have proved to ourselves that the base angles are equal. And if we called the measure of these base angles x, then we know that x plus x plus 90 have to be equal to 180. Or if we subtract 90 from both sides, you get x plus x is equal to 90 or 2x is equal to 90. Or if you divide both sides by 2, you get x is equal to 45 degrees. So a right isosceles triangle can also be called-- and this is the more typical name for it-- it can also be called a 45-45-90 triangle. And what I want to do this video is come up with the ratios for the sides of a 45-45-90 triangle, just like we did for a 30-60-90 triangle. And this one's actually more straightforward. Because in a 45-45-90 triangle, if we call one of the legs x, the other leg is also going to be x. And then we can use the Pythagorean Theorem to figure out the length of the hypotenuse. So the length of the hypotenuse, let's call that c. So we get x squared plus x squared. That's the square of length of both of the legs. So when we sum those up, that's going to have to be equal to c squared. This is just straight out of the Pythagorean theorem. So we get 2x squared is equal to c squared. We can take the principal root of both sides of that. I wanted to just change it to yellow. Last, take the principal root of both sides of that. The left-hand side you get, principal root of 2 is just square root of 2, and then the principal root of x squared is just going to be x. So you're going to have x times the square root of 2 is equal to c. So if you have a right isosceles triangle, whatever the two legs are, they're going to have the same length. That's why it's isosceles. The hypotenuse is going to be square root of 2 times that. So c is equal to x times the square root of 2. So for example, if you have a triangle that looks like this. Let me draw it a slightly different way. It's good to have to orient ourselves in different ways every time. So if we see a triangle that's 90 degrees, 45 and 45 like this, and you really just have to know two of these angles to know what the other one is going to be, and if I tell you that this side right over here is 3-- I actually don't even have to tell you that this other side's going to be 3. This is an isosceles triangle, so those two legs are going to be the same. And you won't even have to apply the Pythagorean theorem if you know this-- and this is a good one to know-- that the hypotenuse here, the side opposite the 90 degree side, is just going to be square root of 2 times the length of either of the legs. So it's going to be 3 times the square root of 2. So the ratio of the size of the hypotenuse in a 45-45-90 triangle or a right isosceles triangle, the ratio of the sides are one of the legs can be 1. Then the other leg is going to have the same measure, the same length, and then the hypotenuse is going to be square root of 2 times either of those. 1 to 1, 2 square root of 2. So this is 45-45-90. That's the ratios. And just as a review, if you have a 30-60-90, the ratios were 1 to square root of 3 to 2. And now we'll apply this in a bunch of problems.