Special Right Triangles One of the good things about math is that you can recreate it yourself, if you can remember the basics. So let’s pretend you suddenly have a Special Right Triangles test, but only vaguely remember anything about them.

Special Right Triangles YIKES! Don’t Panic.

Special Right Triangles Looking carefully, I see there are only two kinds of right triangles here….

Special Right Triangles There is the 30-60-90…. … and the 45-45-90.

Special Right Triangles Let’s deal with this one first... 1 1 And instead of dealing with x, let’s make it easier and have the length of the legs be 1.

Special Right Triangles Ok… so, it’s a right triangle… and the first thing I think of when I see a right triangle is….. 1 1 THE PYTHAGOREAN THEOREM!

Special Right Triangles c = sqr root 2 …and if I want the hypotenuse, all I have to do is solve 12 + 12 = c2. 1 1

Special Right Triangles sqr root 2 • r …and every triangle that has the same angles as this one will be similar to it… 1 • r 1 1 • r 1 …which means that they will all be dilations of this one… with some zoom factor/ratio that I can call r.

Special Right Triangles sqr root 2 • r OK… Now I’m ready… bring on the problems. 1 • r 1 1 • r 1

Special Right Triangles sqr root 2 • r This one is 45-45-90. The length of one leg is 18… which means 18 = 1 • r. 1 • r 1 1 • r 1 So it’s easy enough to figure out that 18 = r. And since the hypotenuse is r • sqr root 2…

Special Right Triangles sqr root 2 • r x = 18 • sqr root 2 1 • r 1 1 • r 1

Special Right Triangles sqr root 2 • r NEXT! This one is also 45-45-90. 1 • r 1 1 • r 1 In fact, the only difference is that r = 3 • sqr root 2 And since the hypotenuse is r • sqr root 2…

Special Right Triangles sqr root 2 • r 1 • r 1 1 • r 1 x = (3 • sqr root 2) • sqr root 2 x = 3 • (sqr root 2 • sqr root 2) x = 3 • 2 x = 6

Special Right Triangles sqr root 2 • r NEXT! This one is also 45-45-90. 1 • r 1 1 • r 1 But we’re given the hypotenuse, instead of a leg! We know the hypotenuse is r • sqr root 2…

Special Right Triangles sqr root 2 • r 1 • r 1 1 • r 1 18 = r • sqr root 2 18 • sqr root 2 = r • sqr root 2 • sqr root 2 18 • sqr root 2 = r • 2 9 • sqr root 2 = r … and so does x

Special Right Triangles 600 600 600 Let’s take on the 30-60-90 now. This one starts off as an equilateral triangle… with all sides equal… and all angles equal to 60 degrees. Then, we cut it in half.

Special Right Triangles 300 300 2 2 600 600 1 1 So now, the two angles at the top are 30 degrees each. And if the original sides of the equilateral triangle had a length of two, the bottom is cut in half, too!

Special Right Triangles 300 2 600 1 Now, let’s just look at the half we care about… the 30-60-90 triangle. Notice that the hypotenuse is twice as long as the side opposite the 300 angle. That’s always going to be true!

Special Right Triangles 300 2 h2 = 3 h h = sqr root 3 600 1 What about the height? This is a job for….. THE PYTHAGOREAN THEOREM! a2 + b2 = c2

Special Right Triangles 300 2 • r r • sqr root 3 600 1 • r Because every 30-60-90 triangle will be similar to this one… The sides will always be proportional to these sides! So we are all set to get started.

Special Right Triangles 300 300 2 • r r • sqr root 3 600 1 • r The missing angle is 300. We are given the length of the side opposite that angle, so r = 8. The hypotenuse, y, is equal to 2r… or 16. The side across from the 600 angle has to be r • sqr root 3… so x = 8 • sqr root 3

Special Right Triangles 300 2 • r r • sqr root 3 600 1 • r Let’s do another.

Special Right Triangles 300 2 • r r • sqr root 3 600 1 • r The hypotenuse, which has to be 2 • r, is equal to 11. That means r, the side opposite the 300 angle, has to be 5.5…. and so x = 5.5. The side across from the 600 angle has to be r • sqr root 3… so y = 5.5 • sqr root 3

Special Right Triangles 300 2 • r r • sqr root 3 600 1 • r Bring on the next one!

Special Right Triangles 300 2 • r r • sqr root 3 300 600 600 1 • r Since this is an isoceles triangle, the other base angle is also 600. And the half-angle on the right is 300. And we can focus on just the part we care about!

Special Right Triangles 300 2 • r r • sqr root 3 300 600 600 1 • r The hypotenuse, which has to be 2 • r, is equal to 20. That means r, the side opposite the 300 angle, has to be 10…. and so y = 10. The side across from the 600 angle has to be r • sqr root 3… so x = 10 • sqr root 3

Special Right Triangles 300 2 • r r • sqr root 3 600 1 • r And, finally….

Special Right Triangles 300 2 • r r • sqr root 3 600 1 • r This time, we are given the length of the side opposite the 600 angle, which has to be r • sqr root 3. If 12 = r • sqr root 3… 12 • sqr root 3 = (r • sqr root 3) • sqr root 3 12 • sqr root 3 = r • (sqr root 3 • sqr root 3) 12 • sqr root 3 = r • 3 4 • sqr root 3 = r

Special Right Triangles 300 2 • r r • sqr root 3 600 1 • r Since r = 4 • sqr root 3… and that is the side opposite the 300 angle… x = 4 • sqr root 3

Special Right Triangles 300 2 • r r • sqr root 3 600 1 • r And, again, since r = 4 • sqr root 3… and the hypotenuse (y) has to be twice as long… y = 8 • sqr root 3

Special Right Triangles sqr root 2 300 2 • r r • sqr root 3 1 • r 1 • r 600 1 • r So, if you ever have to answer questions about Special Right Triangles, now you know that you can create the “formulas” from scratch, just by using the Pythagorean Theorem.