To follow this image, view from right to left starting with what we were given first, an equilateral triangle with sides all equaling 1. Since it's an equilateral triangle, we will also have all angles equal to 60°. We cut the triangle down it's altitude because we want to have right triangles, in this case we get two of them but notice that one of the angles is now 30°, one of the sides is unknown, which happens to be the longer leg, and the shorter leg has been halved to 1/2.
To find the unknown side, we use the Pythagorean theorem but instead of using 1/2 we should double it so that we can have an easier time solving for it. We can do this because the triangle would still hold proportionate to a 30-60-90 as long as we double the other known side. We find the unknown side to be √3 and we set up our values into a special ratio for our special right triangle by putting n into the values so it can fit any 30-60-90 triangle as long as it fits the ratio.
Once again start with what we were given, this time a square with all sides equaling 1 and the angles each being 90°. We cut across this square to get our two right triangles, both having sides equaling 1 and having two 45° angles but now their hypotenuse has an unknown value. To find this unknown value you use the Pythagorean Theorem and you get √2. We have our final values and turn them into the special ratio for our 45-45-90 triangle by using n as a variable for any other special right triangles we may have to make sure the sides correspond to the ratio of 1, 1, and radical 2.
Inquiry Activity Reflection
1. Something I never noticed before about special right triangles is that they can be constructed from other shapes. To go into more detail I just thought they were weird triangles that had special rules attached to them but they actually appear in a lot of places.
2. Being able to derive these patterns myself aids in my learning because it gives me an idea of where these special rules come from and how they come together.