ID#2: Unit O Concepts 7-8: How can we derive the patterns for our special right triangles?

Inquiry Activity Summary:

In the activity we did during class, we were supposed to derive the special right triangles making all the sides equal to one. To derive a 45-45-90 triangle, we were given a square with each side equal to 1. From there, we had to divide the square into two parts so that we can have a 45-45-90 triangle and then find the value of the hypotenuse. To derive a 30-60-90 triangle, we were given an equilateral triangle with side lengths equal to 1. We had to again cut the triangle in two parts to get the triangle degrees of 30-60-90. Finally,we found the values of the side missing.


1. When deriving the equilateral triangle, you must cut the triangle in half so that one side can remain as 60 degrees, one can be 90 degrees, and the other one can be 30 degrees since it is half of the triangle. We know that all sides of the equilateral are equal to 1. Since it is cut in half, the bottom side is equal to 1/2. (as shown on the bottom left) The hypotenuse is equal to 1, so we know what a & c are equal to. So to find the value of the height, we must use the pythagorean theorem. My work is shown on the bottom right which gives us the value of the height which is rad 3/2. We know that 

 a 30-60-90 triangle has side values of n, n rad 3, and 2n. The variable "n" is a constant. In this case, our n is 1/2 which gives us these answers but the value of n can vary depending on the problem they give you. But, they all end up falling under the same guidelines. 

2. When deriving a 45-45-90 triangle by an equilateral triangle we must first cut the square diagonally. It must be diagonally because it will create two triangles and also making two 45 degree sides. It can't be cut in the middle because it will make a rectangle. We know that two sides are equal to 1 so we must use the pythagorean theorem, as shown below, to find the value of the hypotenuse. We use "n" to represent our sides because it is a constant. This means that any number will follow the same guidelines for each side we must know. 

Inquiry Activity Reflection: 

1. Something I never noticed before about special right triangles is that the value of the sides are directly connected to equilateral triangles and squares. The value of their sides make sense when you work out the problem to its individual parts.

2. Being able to derive these patterns myself aids in my learning because I finally understand where the values of the sides for special right triangles come from and I am able to see how things are directly connected to one another.