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

INQUIRY ACTIVITY SUMMARY: 

http://www.proprofs.com/flashcards/upload/a3979269.jpg

In the activity I derived a special right triangle know as an equilateral triangle(shown on the right). The goal was to derive a 30-60-90 triangle from and that could be done by bisecting the triangle in half(like in the picture shown below). Thus creating a 30-60-90 triangle.

http://bitmote.com/public/tetrahedron/.halfEquiTriangle3_s.jpg

In the activity, I was given an equilateral triangle with the lengths of 1, which I bisected in half and received 1/2 as the bottom length. So I knew that my hypotenuse was one and my bottom leg was 1/2, and by using Pythagorean Theorem which is a^2 + b^2 = c^2. I used 1/2 as my a value and 1 as my c value because I was looking for b. Check the picture below to see my work process and I got radical 3 over 2 as the value of b. I then distinguished a pattern among the 30-60-90 triangle. I noticed that 1/2 could be viewed as "n" and 1 as "2n" and radical 3 over 2 as "n rad 3". This would work with any numbers because that is the relationship that all the side have.       

http://blog.powerscore.com/Portals/156640/images/satblog-5.jpg

On the other side of the worksheet, I had to do the same thing but with a 45-45-90 triangle that can be derived from a square. The first thing that I did was draw a line across the square, from one 90 degree angle to the other. Like the picture to the left. I was told that my side lengths for the square were 1. I knew that would have to use Pythagorean Theorem again so that I could solve for my hypotenuse. If you see the bottom picture to the left you'll see my work process for the Pythagorean Theorem. As I looked at the triangles I saw there was a pattern. The 1's could be viewed as n and rad 2 could be viewed as n rad 2 because of the relationship the sides have.

  


 
1. In the picture on the left, it shows how to derive a 30-60-90 triangle from a equilateral. Make sure to keep note on what I mention about the patterns I used Pythagorean Theorem to find my b value and after i found the patter that 1 could be 2n and 1/2 could be n and rad 3 over 2 could be n rad 3.  This pattern works for any example check out the other picture to see what I mean.

2. In the following picture, I demonstrate how a 45-45-90 triangle can be derived from a square. Since my side lengths are 10 and I basically cut the square diagonally I bisected the 90 degree angles and made them into 45 degree angles. I knew that i had legs with the length of 10, so I used Pythagorean Theorem to help me find the hypotenuse which was 10 rad 2.  If you look at my work i originally got rad 200 but I broke that into 20 and 10, then broke down the 20 into 2 and 10 which gave me 10 rad 2. The pattern for a 45-45-90 triangle is that since the length would be the same length because it originates from a square you can use those values as "n" and the hypotenuse could be n rad 2. This pattern works for every example, Don't believe me ?  Check out the picture below.













HEADING FOR THIS SECTION: INQUIRY ACTIVITY REFLECTION 

1. “Something I never noticed before about special right triangles is…” they were found in squares such as the 45-45-90. I knew that the 30-60-90 triangle could be found in an equilateral triangle. 
   
   
   
   
   
   
   2. “Being able to derive these patterns myself aids in my learning because…”
   I get a better understanding on where the patterns actually come from instead of just memorizing them.