1. For a 30-60-90 triangle, we must cut it straight down because it was currently an equilateral triangle, meaning all the angles are the same. Since we cut it in half, the length will be cut in half as well, making it 1/2. Since it was given that the hypotenuse side is one and we found out that the length is 1/2, we just need to find the height. To find the height, you use the pythagorean theorem which is a^2+b^2=c^2 (look at picture for mathematical explanation). To get rid of the fraction, you multiply everything by two. We use "n" as if it were equal to one. As a result, the hypotenuse is 2n, the side across the 60 degrees angle is n radical 3 and the last side is n. We use n because it's a variable that represents 1 since it's in all the terms.
2. We can derive a 45-45-90 triangle from a square by cutting it diagonally. Cutting it diagonally will split the square in half as well as the 90 degree angles, making it 45 degrees. We know that the sides opposite from the 45 degree angles are still 1 because we didn't cut any sides in half, just the angles. To get the hypotenuse, you use the pythagorean theorem. (1)^2 + (1)^2 = c^2, C should equal radical 2 according to the work shown below. Also, we use n as a constant or a variable which represents 1. Using "n" represents the relationship between the sides and angles.
Inquiry Activity Reflection:
1. "Something I never noticed before about special right triangles is..." That there is a reason why the sides are the way they are, it's not just memorization.
2. "Being able to derive these patterns myself aids my learning because..." There is logical reasoning in deriving the patterns and I won't have to memorize it like I did before.