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# Special right triangles intro (part 2)

## Video transcript

sorry for starting the presentation with a cough I think I still have a little bit of a bug going around but now I want to continue with the 45-45-90 triangles so in the last presentation we learned that either side of a 45-45-90 triangle that isn't the ipod news is equal to the square root of 2 over 2 times the hypotenuse let's do a couple of more problems so if I were to tell you that the hypotenuse of this triangle and it's a 45 once again this only works for 45-45-90 triangles and if I just draw one 45 you know the other angles got to be 45 as well if I told you that the hypotenuse here is mm let me say 10 all right we know this is hypotenuse because it's opposite the right angle and then I want to ask you what this side is X well we know that X is equal to square root of 2 over 2 times the hypotenuse so that's square root of 2 over 2 times 10 or X is equal to 10 and whoops is equal to 5 square roots of 2 right 10 divided by 2 so X is equal to 5 square root of 2 and we know that this side and this side are equal right I guess we notice it's an isosceles triangle because these two angles are the same so we also know that this side is 5 over 2 and if you're not sure try it out let's try the Pythagorean theorem we know from the Pythagorean theorem that 5 root 2 squared plus 5 root 2 squared is equal to the hypotenuse squared the hypotenuse is 10 is equal to 100 well this is just 25 times 2 so that's 50 plus 25 times 2 is 50 this is 100 up here is equal to 100 and we know of course that this is true so it worked we proved it using the Pythagorean theorem and that's how actually we came up with this formula in the first place if you want to go back to one of those presentations if you forget how we came up with this I'm actually now going to introduce another type of triangle and I'm going to do it the same way by just posing a problem to you and then using the Pythagorean theorem to figure it out this is another type of triangle called a 30-60-90 triangle 30-60-90 triangle if I don't have time for this I will do another presentation let's say I have a right triangle that's not a pretty one but we'll use what we have that's the right angle and if I were to told you tell you that this is a 30-degree angle well we know that the angles in a triangle app to add up to 180 so this is 30 this is 90 and let's say that this is X X plus 30 plus 90 is equal to 180 because the angles in a triangle add up to 180 we know that X is equal to 60 right this angle is 60 and this is why it's called a 30-60-90 triangle because that's the names of the three angles in the Triangle and if I were to tell you that the hypotenuse is let's say the hypotenuse of this triangle let me instead of calling it C like we always do it let's call it H and I want to figure out the other sides how do we do that well we can do that using pretty much the Pythagorean theorem and here I'm going to a little trick let's draw another copy of this triangle but draw it exactly flip it over drawing it draw it the other side and this is the same triangle it's just faced in the other direction right if this is 90 degrees we know these two angles are supplementary you might want to review the angles module if you forget that two angles that share kind of this common line would add up to 180 degrees this is 90 this will also be 90 and you could eyeball it it makes sense and since we flipped it this triangle is the exact same triangle is this it just flipped over the other side we also know that this angle is 30 degrees and we also know that this angle is 60 degrees right well if this angle is 30 degrees in this angle is 30 degrees we also know that this larger angle goes all the way from here to here it is sixty degrees right well if this angle is 60 degrees this top angle is 60 degrees and this angle on the right is 60 degrees then we know well we know that we know from the theorem that we learned when we did 45-45-90 triangles that if these two angles are the same then the sides that they don't share have to be the same as well so what are the sides they don't share well it's this side on this side so if this side is H then this side is H right but this angle is also 60 degrees so if we look at this 60 degrees and this 60 degrees we know that the sides that they don't share are also equal well they share this side so the sides that they don't share are this side and this side so this side is H we also know that this side is H right so it turns out that if you have 60 degrees 60 degrees in 60 degrees that all the sides are the same length or it's an equilateral triangle and that's some just something to keep in mind and that makes sense too because an equilateral triangle is symmetric no matter how you look at it so it makes sense that all the angles would be the same and all of the sides would have the same length but when we when we originally did this problem we only used half of this equilateral triangle so we know this whole side right here is of length H but if that whole side is of length H well then this side right here just the base of our original triangle and I'm trying to be messy on purpose let me try it in another color this is going to be half of that side right because that's H over 2 and then this is also H over 2 right over here H over 2 so if we go back to our original triangle and we said this is 30 degrees and that this is the hypotenuse because it's opposite the right angle we know that the side opposite the 30-degree side is 1/2 of the hypotenuse and just to reminder how did we do that well we double the triangle turned into an equilateral triangle figured out this whole side has to be the same size but anews and this is half of that old sign so it's half of the hypotenuse so let's remember that the side opposite the 30-degree side is 1/2 of the hypotenuse let me redraw that on another page because I think this is getting messy so going back to what I had originally this is the right angle this is the hypotenuse this side right here if this is 30 degrees we just derived that the side opposite the 30 degrees the side opposite right it's like what the angle is opening into that this is equal to 1/2 the hypotenuse if this is equal to 1/2 the hypotenuse then what is this side equal to well here we can use the Pythagorean theorem again we know that this side squared plus this side squared let's call this side a is equal to h squared so we have 1/2 whoops 1/2 H squared plus a squared is equal to h squared this is equal to h squared over 4 plus a squared is equal to h squared well we subtract h squared from both sides we get a squared is equal to h squared minus h squared over 4 so this equals a squared times 1 minus 1/4 this is equal to 3/4 h squared excuse the call once again that's equal to a squared I'm running out of space so I'm going to go all the way over here so take the square root of both sides and we get a is equal to the square root of 3/4 is the same thing as the square root of 3 over 2 and then the square root of H is H squared just H and this a remember this is an area this is what we decided the length of the side I probably shouldn't have used a but this is equal to the square root of 3 over 2 times H so we're there we've derived what all the sides relative to the hypotenuse are of a 30-60-90 triangles this is a 60-degree side so if we know the hypotenuse and we notice a 30-60-90 triangle we know the side opposite the 30-degree side is 1/2 the hypotenuse and we know the side opposite the 60-degree side is a square root of 3 over 2 times the hypotenuse in the next module I'll show you how using this information which you may or may not want to memorize it's probably good to memorize and practice with because it'll make you very fast on standardized tests how we can use this information to to solve the sides of a 30-60-90 triangle very quickly see you in the next presentation