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

Inquiry activity summary:

 
1. 30-60-90 We begin by taking a regular triangle which is 60 in each of its corners because the sum of a triangle is 180 degrees. We split a line down the triangle in order to achieve our special right triangle which is a 30 60 90 triangle. When we split the triangle we make a 90 degree angle and split the 60 in half which changes the value to 30 and our 60 in the corner remains. Since we split the triangle in half our one becomes 1/2 for each side because 1divided by 2 is one half.  Our base of our triangle is labeled as A while the side is labeled B and the hypotenuse as C.We then take Pythagorean theorem which is a^2+b^2=c^2. Our base is labeled as A, our side is labeled as B, and our hypotenuse is labeled as C. We then plug in and square our values.  The product results in b= radical 3 over 4 in which we square the 4 and leave the radical 3. This value of radical 3 over 2 is across from our 60 degree angle. Since we have fractions, and a majority of people despise fractions we multiply each value by 2 so radical 3 over 2 becomes simply radical 3 across from our 60 degree angle. Our 30 degree angle which was once 1/2 becomes simply one. Our hypotenuse becomes then 2.  When our values vary we use a variable like n for extended values so we can use them for the numerous amount of numbers.
 

 



 
2. We are given a square with equal sides of one which has the angles of 90 degrees in 4 spots. Taking our square, we split  it diagonally so we can get our 45 45 90 triangle. With two 90 degree angles cut in half  we get the result of 2 45 degree angles and have a 90 degree angle remaining.We start off using the Pythagorean theorem  with a^2+b^2=c^2.  Our A is labeled at the base our B is the side and our hypotenuse is C. We plug in our value of one which results in c equals radical 2 which is our hypotenuse. We put N on the side just in case our numbers change.

Inquiry activity reflection:
1. Something I never noticed before about right triangles is how you can derive them from other shapes like a whole square or a whole triangle.
2. Being able to derive these patterns myself aid in my learning because I can use our knowledge of shapes and break down and use for knowledge later on.