I/D2: Unit O - How can we derive the patterns for our special right triangles?

Inquiry Activity Summary

1) 30-60-90 Triangle

In order to derive the 30-60-90 from an equilateral triangle whose sides all equal 1 (in this case), we start out by cutting it straight down the middle (vertically) in a way that the triangle becomes two identical triangles. Original, the triangle is made up of all 60 degree angles. By cutting it in half, the top angle become 30 degrees, and either the right or left angle (depending on which one you use) is the 60 degree angle, thus giving you the 30-60-90 Triangle.

In the last three pictures, you can see the way that the 30-60-90 triangle is formed. We know that each leg should be equal to 1, but we have one side that has to values of 1/2 when it is split. This is simply to show that when we split the triangle down the middle, you end up with 2 separate 30-60-90 triangles with legs being equal to 1, 1/2, and the b leg is equal to radical 3/2. The way you find the value of leg b is shown in the next picture.

Here, we have the Pythagorean Theorem. As you can see, we have 1/2 to the second power. This gives us 1/4, which we subtract from 1 on the other side. we are left with radical 3/4 which then simplifies to radical 3 divided by 2. 

Here, we have the 30-60-90 triangle, but we have placed "n" in each value to show that there is a proportion in the triangle. The values of each leg form a ratio, which remains the same regardless of what value you use for n. Since you multiply fairly, you multiply the same amount for "n", thus ending up with the same ratio you started with.

2) 45-45-90 triangle

In order to obtain this triangle from a square, we have to cut the square diagonally. Since we start out with a square, all sides are equal to 1 (the given value), and we simply need to find the value of the hypotenuse.

In these two last pictures, we see the transformation from the square to the triangle. Using what we know of the 45-45-90 triangle, we know that by cutting the square diagonally, your 90 degree angles in two of the corners will split in half to give you the 45 degree angles that we need. Then, we know that the legs directly opposite to the 45 degree angles are "n", which means that n=1 (since legs=1). We know that the leg opposite to the 90 degree angle is n radical 2, which means that the hypotenuse is radical 2.

This picture goes over the way to get the hypotenuse using Pythagorean Theorem. You get the same answer, but have a couple extra steps.

This picture shows the original square, with substituted into the values. Like stated in the 30-60-90, we use the letter "n" to show the ratio between all the legs, and just like in the 30-60-90 triangle, this ratio is the same regardless of what value you substitute for "n".

Inquiry Activity Reflection

Something I never noticed before about special right triangles is that you can derive them from either a square or a triangle of same values.

Being able to derive these patterns myself aids in my learning because I can quickly derive the Special Right Triangles during a test if I forget all else, and possibly pass the test.