okay 8-2 is about special right triangles okay they're special because they're used more than other kinds of right triangles okay the first kinds if you take a square and you cut it in half diagonally um you get a right triangle okay the angles created is 45 45 and 90 and so it's called a 45 4590 right triangle okay now because it's an isoceles triangle and also because it's half of a square um these two legs have the exact same length so if this one was five this one would be five okay if this one was seven this would be seven see these two legs have the exact same length in a 45 45 90 right triangle so here's a couple of examples um now I'm I'm going to use the Pythagorean theorem to figure out what the length of the hypotenuse is are uh but my goal is to find a faster shortcut instead of doing this every time but um off this is X and this these are all right triangles so the Pythagorean theorem works so 2 s + 2^2 is = to x^2 x being the hypotenuse so 4 + 4 is = x² and X2 is equal to 8 and so X must be equal to the square < TK of 8 um now I'm making a factor tree 8 becomes two and four um two and two I have my pair a perfect square so X is equal to 2 < tk2 2 < tk2 okay me move this out of my way okay um five and five again I'll call this x um 5^ 2 + 5^2 is equal to X2 um 25 + 25 is = x^2 25 and 25 make 50 so x equal to the square root of 50 to simplify that again I'll make a factor tree 50 let's say 2 and 25 25 is 5 * 5 I have a perfect square so X is equal to 5un um two 5 < tk2 now um if we did this last one we would get one < tk2 so if you'll notice there's a pattern of if if the leg is two the hypotenuse is 2 < tk2 if the leg is five the hypotenuse is 5 < tk2 if the leg is one it's one < tk2 there's a pattern in the textbook it textbook puts it this way um hypot in a 45 4590 triangle both legs are congruent and the length of the hypotenuse is < tk2 * the length of a leg I think this um equation is pretty helpful to us okay an example um since the leg is seven I just saw the pattern was that this must be 7 < tk2 again you can use the Pythagorean theorem but that's what it's going to be and this one where the hypotenuse is eight it's kind of harder to figure out what x must be I'm going to turn to my formula of um hypotenuse is equal to 8 I'm sorry is sare two time leg the hypotenuse is 8 S2 * the leg is X let's see here to get X by itself I divide both sides by the square of two okay so x equal to 8 over the sare of two now unfortunately this isn't in simplest form so the way I simplify is multiply top and bottom by the square of two which is what the denominator is and 8 * < tk2 is 8 < tk2 / < tk2 * < tk2 isun 4 now I know the square of 4 is equal to 2 and 8 over2 simplifies to be 4 over one so this equal to 4 < tk2 okay now there is a shortcut to this as well if you remember it in a 45 4590 right triangle um you take the hypotenuse and divid by two and get four this four time the square two okay so if you take half of the hypotenuse and multiply by the sare of two um that will work that'll be equal to the legs here's a couple of examples now just to check your work um H will be equal to 9 < tk2 x will be equal to 4 um this x will be equal to 3 < tk2 and X is equal to 5 < tk2 moving on the other kind of special right triangle is half of an equilateral triangle which if you cut it in half you get a 30 60 90 right triangle the important thing about the side lengths here is that since we've cut it in half this leg is half of the hypotenuse okay so here's a couple of examples if this shortest leg is four then the hypotenuse must be eight since it's double if the hypoten is 12 the shortest leg must be six and this if the shorter leg is is three the hypotenuse must be six okay again this is a right triangle we can use the Pythagorean theorem 4^2 + x s must be equal to 8^ 2 42 16 + x^2 is equal to 64 that's here solving for x I'll subtract 16 from both sides X2 is equal to 48 and X is equal to the square root of 48 so to simplify this I'll use a factor tree let's see here 4 * 12 4 is 2 * 2 and 12 is 3 * 4 and four is two and two so I have I pair and a pair here which gives me perfect squares 2 * 2 < tk3 is 4 < tk3 sorry it's all jumbled um group some things here okay um now the pattern if we did the other ones as well the same way here we get 6 < tk3 here we get 3 < tk3 you'll notice there's a pattern of whatever this shortest length is that times root3 is what the longer leg is 6 * root3 is 6 Ro tk33 3 * 3 is um 3 tk3 that's the pattern the textbook puts it this way again I think that these two equations are pretty helpful to us the hypotenuse is equal to two times the shorter leg and the longer leg is equal to the root3 times the shorter leg okay couple of examples that side doesn't really matter okay that side doesn't really matter matter so let's see here if the hypotenuse is 12 the shortest side must be half of that or six okay and the longer leg is the shorter leg times < tk3 so X must be equal to 6 < tk3 and I'm done in this one the hypotenuse is six the shorter leg is well half of six so X must be three which by the way this longer leg um would be three < tk3 all right couple more examples if the hypotenuse is 40 the the shorter leg will be half of that so half of 40 is 20 and the longer leg is the shorter leg times the square otk of three so it's 20 < tk3 number two the hypotenuse is 10 the shorter leg is half of 10 or five and the longer leg is 5 * the < TK of 3 okay number three is a little more interesting the longer leg is 2 < tk3 so um y the shortest length times the sare of 3 is equal to 2 < tk3 so y must be equal to two so the 2 * root3 is 2 < tk3 so if the shorter length is two um the hypotenuse is double that and X must be equal to four okay few more practice problems we don't have time to do this right now um okay one last type of question I want to talk about is the side lengths of a triangle are given determine if the triangle is a 45 4590 triangle a 3060 triangle or neither um this doesn't have any doubling going on any Pairs and um this doesn't look like it's going to be 40 < tk3 so this must be neither number eight um although I have a double here just like a 30 6090 triangle um 30 6090 triangles have root3 not < tk2 so this must be neither as well in number six oh these two are the same and I have a root2 um pattern going and so this must be a 45 45 90 triangle number 10 oh I have doubling just like a 30 6090 and have aunk three and so this must be a 30 60 90 triangle