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# Special right triangles intro (part 1)

## Video transcript

welcome to the presentation on 45-45-90 triangles let me write that down oh I'll come to pity so there you go 45-45-90 triangles or we could say 45-45-90 right triangles but that might be redundant because we know any angle that has a ninety degree measure in it is a right triangle and as you can imagine the 45-45-90 these are actually the degrees of the angles of the triangle so why are these triangles special well if you saw the last presentation I gave you a little theorem that told you that if two of a base angles of a triangle are equal and I guess only a base angle if you draw it like this you could draw it like this in which cases maybe not so obviously a base angle but it would still be true if these two angles are equal then the size that they don't share so this side on this side in this example or this side and this side in this example that the two sides are going to be equal so what's interesting about a 45-45-90 triangle is that it is a right triangle that has this property and how do we know that it's the only right triangle that has this property well you could imagine a world where I told you that well this is a right triangle this is 90 degrees so this is the hypotenuse right so sides opposite the 90 degree angle and if I were to tell you that these two angles are equal to each other what do those two angles have to be well if we call these two angles X we know that the angles in a triangle add up to 180 so we'd say X plus X plus this is 90 plus 90 is equal to 180 or 2x plus 90 is equal to 180 or 2x is equal to 90 or X is equal to 45 degrees so the only right triangle in which case in which the other two angles are equal is a 45 45 90 triangle so what's interesting about a 45-45-90 triangle well other than what I just told you let me I want to be redrawn I'll be draw it like this so we already know that this is 90 degrees this is 45 degrees this is 45 degrees and we based on what I had just told you we also know that the sides that the 45 degree angles don't share are equal so this side is equal to this side and if we're viewing it from a Pythagorean theorem point of view this tells us that the two sides that are not the hypotenuse are equal so this is a hypotenuse so let's call this side a decide B so we know the from the Pythagorean theorem let's say about hypotenuse is equal to C the Pythagorean theorem tells us that a squared plus B squared is equal to C squared right excuse me well we know that a equals B because this is a 45-45-90 triangle so we could substitute a for b or b for a but let's just substitute b for a so we could say b squared plus b squared is equal to C squared or 2b squared is equal to C squared or B squared is equal to C squared over 2 or B is equal to the square root of C squared over 2 which is equal to C because we just take the square root of the numerator and the square root of the denominator C over the square root of 2 and actually even though this is a topic on this this is a presentation on triangles I'm going to give you a little bit of extra information on something called rationalizing denominators so this is perfectly correct we just derived that B and we also know that a equals B but B is equal to C divided by the square root of 2 it turns out that in most of mathematics and I never understood quite exactly why this was the case people don't like square root of two's in the denominator or in general they don't like irrational numbers in the denominator irrational numbers are numbers that you know they have decimal places that never repeat never end so the way that they get rid of irrational numbers and the denominators that you call it you do something called rationalizing the denominator in the way you rationalize the denominator let's take our example right now if we had C over the square root of 2 we just multiply both the numerator and the denominator but the same number right because you multiply something numerator and the denominator by the same number that's just like multiplying it by 1 and square root of 2 over square root of 2 is 1 and as you see we reason why we're doing this is because square root of 2 times square root of 2 what's the square root of 2 times square root of 2 right it's 2 right we just said I mean something times something is 2 well the square root of 2 times square 2 that's going to be 2 and then the numerator is C times the square root of 2 so notice C times the square root of 2 over 2 is the same thing as C over the square root of 2 and this is important to realize because sometimes while you're taking a standardized test or you're doing a test in class you might get an answer that looks like this has a square root of 2 or maybe even a square root of 3 or squared whatever in the denominator and you might not see your answers if it's a multiple choice question what you need to do in that case is rationalize the denominator so multiply the numerator the denominator by square root of 2 and you'll get you know square root of 2 over 2 but anyway back to the problem so what did we learn this is equal to B right so it turns out that B is equal to C times the square root of 2 over 2 so let me write that so we know that a equals B right and that equals the square root of 2 over 2 times C now you might want to memorize this although you can always derive it if you use the Pythagorean theorem and remember that the size that aren't the hypotenuse at a 45-45-90 triangle are equal to each other but this is very good to know because if you're say you're taking the SAT and you just solve a problem really fast and if you have this memorized you and someone gives you the hypotenuse you can figure out one of the sides very fast or someone gives you one of the sides you can figure out the hypotenuse very fast let's try that out so I'm going to erase everything so we learned just now that a is equal to B is equal to the square root of 2 over 2 times C so if I were to give you a right triangle and I were to tell you that this angle is 90 and this angle is 45 and that this side is let's say the side is 8 I want to figure out what this side is well first of all let's figure out what size the hypotenuse well the hypotenuse is the side opposite the right angle so we're trying to actually figure out the hypotenuse let's call the hypotenuse C and we also this is a 45-45-90 triangle right because this angle is 45 so this one also has to be 45 because 45 plus 90 plus 90 is equal to 180 so this is a 45-45-90 triangle and we know one of the sides the side could be a or b we know that eight is equal to the square root of two over two times C C is what we're trying to figure out so if we multiply both sides of this equation by two times the square root of two all right I'm just multiplying it by the inverse of the coefficient on C because then the square root of two cancels out that square root of 2 this 2 cancels out with this 2 we get 2 times 8 16 over the square root of 2 equals C which would be correct but as I just showed you people don't like having radicals in the denominator so we could just say C is equal to 16 over the square root of 2 times the square root of 2 over the square root of 2 so this equals 6 square roots of 2 over 2 which is the same thing as 8 square roots of 2 so C in this example is 8 square roots of 2 and we also know since this is a 45-45-90 triangle that this side is 8 hope that makes sense in the next presentation I'm going to show you a different type of triangle actually I might even start off with a couple of more examples of this because I feel I might have rushed it a bit anyway I'll see you soon in the next presentation