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CCSS.Math:

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 the hypotenuse is X then the shorter side is x over two on the side in between the side that's opposite the 60-degree side is square root of three x over two or another way to think about it is if the shortest side is one I'll do the shorter side then the medium side 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 over 2 but at 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 30 60 or 30 60 90 triangle you could say hey I know how to figure out I know how to figure out one of the sides based on this ratio right over here in just an example if you see a triangle that looks like this where the sides are 2 2 square roots of 3 and 4 once again the ratio of 2 to 2 square roots of 3 is 1/2 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 it is if I have a right triangle that is also isosceles so a right triangle that is also isosceles you obviously can't have a right triangle that is equilateral because an equilateral triangle has all all of their angles have to be 60 degrees but you can have a right angle you couldn't have a right triangle that is isosceles and isosceles let me write this this is a right right isosceles 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 call the measure of these base angles X then we know that X plus X plus 9 you have to be equal to 180 X plus X plus 90 need 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 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 45-45-90 triangle and what I want to do in 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 this was actually more straightforward because in a 45-45-90 triangle if we call each of 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 C so we get x squared x squared plus x squared that's the square of both of the lengths 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 C squared we can take the principal root of both sides of that I wanted to change to yellow and it keeps not letting me okay 2 C squared let us take the principal root of both sides of that principal root of both sides 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 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 this is 90 degrees 45 and 45 like this and you really just have to know two of these angles to know that the other one 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 is 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 the legs so it's going to be 3 times the square root of 2 so the ratio of the sides and hypotenuse in a in a 45-45-90 triangle or a right isosceles triangle the ratio of the sides are one of the legs can be one then the other legs going to have the same measure the same length and then the hypotenuse is going to be square root of two times either of those 1 to 1/2 square root of 2 so this is 45-45-90 let me write this is 45-45-90 that's the ratios and just as a review if you have a 30-60-90 the ratios were 1/2 square root of 3 to 2 and now we'll apply this in a bunch of problems