I/D #2: Unit O - How can we derive the patterns for our special right triangles?

INQUIRY ACTIVITY SUMMARY
1. 30-60-90 Triangle

     The 30-60-90 triangle comes from an equilateral triangle, which has equilateral sides with equiangles of 60 degree. The equilateral triangle is cut in half, making: (2) triangles, (2) 30 degree angles, and (2) 90 degree angle. To explain the relationship between the sides and angles, an equilateral triangle with the sides of 1 can be used. If the process would be applied to this example, and focusing on ONE of the two triangle, the angles would remain the same as mentioned. The side cut in half will equal to 1/2, while the other untouched side (the hypotenuse in this case) would remain as 1. However, one side remains missing with its  value, and in order to find it the Pythagorean Theorem (a^2 + b^2 = c^2) is used.

     Using the Pythagorean Theorem, c would be the hypotenuse while a and b would be the other two sides that are across the 30 and 60 degree angles. Side a would equal to 1/2 while c would equal to 1. Plug those into the equation to solve for b will result with b equaling to radical 3 over 2. Across the 30 degree angle, a= 1/2; across the 60 degree angle, b= radical 3/2; and across the 90 degree angle c= 1. The relationship for a 30-60-90 triangle is 1/2-radical 3/2-1. HOWEVER, not all triangles are the same and not all derive from an equilateral triangle with the side of 1. The variable n can be used on the relation to represent any value for a triangle, creating  (n/2)-n(radical 3)/2-(n). (And to make life simpler, the fraction can be removed my multiplying the relation by 2 to make (n)-n(radical 3)-2n..

2. 45-45-90 Triangle

     The 45-45-90 triangle comes from cutting a square directly in half diagonally. Cutting the square diagonally in half will create: (2) triangles, (4) 45 degree angles, and (2) 90 degree angles. To explain the relationship between the sides and angles, a square with the sides of 1 can be used. Cutting the square and focusing on ONE triangle, the right triangle created will have two sides with the length of 1 with a missing hypotenuse. The sides of 1 will be across the 45 degree angles, and the hypotenuse can be found using the Pythagorean Theorem.

     With the Pythagorean Theorem, side a will be 1 while side b will also be 1. After squaring the sides, adding them, and finding the square root, the hypotenuse will equal to radical 2. So a 45-45-90 triangle has a relationship of 1-1-radical 2. HOWEVER, not all triangles will derive from a square of 1. As in the 30-60-90 triangle, n can be used on the relation to represent any value of the triangle, creating (n)-(n)-n(radical 2).

INQUIRY ACTIVITY REFLECTION
“Something I never noticed before about special right triangles is…” that these special right triangles derived from equilateral triangles with equiangles. The 45-45-90 right triangle came from a square, a quadrilateral with the same length in sides and had (4) 90 right angles. The 30-60-90 right triangle derives from an equilateral triangle that had (3) 60 degree angles.
“Being able to derive these patterns myself aids in my learning because…” it gives me a better understanding where the numbers and relations actually come from and now know that a mathematician didn't randomly chose numbers. Also, in case I forget the relation during an important test, I could remember where and how these patterns are derived from.