1/D: #2 Unit O Concept 7-8: Derive the SRTs

Inquiry Summary Activity:

 To derive the Special Right Triangles, I started with a square with the side lengths of one and a equilateral triangle with the side lengths of 1. Theses two shapes help us derive the 45-45-90 Special Triangle and the 30-60-90 Special Triangle.The Pythagorean Theorem was also used to find missing parts of the triangles.

1) 30-60-90 Triangle:
To derive the 30-60-90 triangle,  I used a equilateral triangle with the side lengths of 1. An equilateral triangle has angles that are 60 degrees and all 3 angles are the same. Since a equilateral triangle has the same angles and the same sides, I sliced it down the middle. By slicing it down the middle a height and a 90 degree angle were created. A 30 degree angle was also created by splitting the 60 degree angle in half. Since the triangle was sliced now the base is 1/2. Since we know that on side is 1 and the base is now 1/2, using the Pythagorean Theorem we can figure what the height is. We use the a^2+b^2=c^2. a=One squared is a=1 and b=1/2 squared is b= 1/4. You add them and that gives radical 3 over 2.We multiply everything by 2 to get radical 3, 2 and 1. This translates into the normal n radical3, 2n, and n. The n is used as variable meaning that any number can be substituted in. N is used to expand the problem as needed.

2) 45-45-90 Triangle:
To derive the 45-45-90 triangle, I used a square with the sides of 1. The 4 angles of a square are 90 degrees.I sliced the square down its diagonal. By cutting it down its diagonal created  a hypotenuse and two 45 degree angles. Since we know that two sides are 1 we must now find the hypotenuse of the triangle created. I used the Pythagorean Theorem to find the hypotenuse. The equation is a^2+b^2=c^2. A=1 and B=1, which means c stays the same. 1 squared is 1 so that means a+b equals one. To get c we must get rid of the square root by squaring c and 2. That means c equals radical 2. The sides of the triangle are 1,1, and radical 2. That translates to the original pattern of a 45-45-90 triangle which is: n,n, n radical 2. The N is used a variable, which means any number can be substituted and that means that N is also used to expand the problem as needed.

Inquiry Activity Reflection:
1) Something I never noticed before about special right triangles is that they were created form other shapes.
2)Being able to derive these patterns myself aids my learning because if I need to do a problem that involves this and the triangles are not given I can do it on my own.