I/D #2: Unit O Concept 7-8: Deriving Patterns for Special Right Triangles

Inquiry Activity Summary

The purpose of the activity was to help us apprehend the special right triangle and how its patterns gets what it has. With the special right triangle, it helps us understand where the patterns come from, and instead of memorizing the patterns for both the 45-45-90 degree triangle and 30-60-90 degree triangle, you see how we get each of their side lengths by deriving them.

1. 30-60-90 Triangle

An equilateral triangle makes up of 60 degrees for each of its 3 sides; and making each of its sides equal to one since all its sides and degrees are equal. But when cut in half, its degrees change, and so does its sides. Its degrees become 30-60-90 degrees; the number across 90 degrees stays the same since it side was not cut off, the number across 30 degrees is now “1/2” because 1 was cut in half, so it value was also cut in half, so know the only thing we know need to find is the number that is across 60 degrees.

We use the Pythagorean Theorem to solve for the number side across 60 degrees, which we then get √3/2 as the answer. We then see how each side effects and correlates to the other sides. To get rid of the fractions from each sides, we multiply each side by 2, which gets rid of the fractions; we then get n, n√3, and 2n. After doing that, we multiply each side by “n”. Since “n” is a ratio, it helps the sides get bigger but its value will always stay constant. 

2. 45-45-90 Triangle

When we cut a square that has its four sides equal to 90 degrees and making each of its sides equal to 1 will change when cutting it across diagonally. It cuts the 90 degree angles in half, making it into a 45-45-90 degree triangles. Since two sides of the now triangle stood the same, it is still equal to one, but we now need to find the other side that was cut in half, so we will now use the Pythagorean Theorem to find the hypotenuse, which is the missing side. After finding the hypotenuse side, √2, we put everything together. We then see how each sides correlate and effect each other.

Since there are two 45 degree angles, both of their sides now equal to 1; and the 90 degree angle equals to √2. We then multiply “n” to all three sides and we end up with n, n, and n√2. Since “n” is a ratio, it always stay constant, so even if the numbers you plug in to “n” make the sides bigger, its value will stay constant.

Inquiry Activity Reflection

1. Something I never noticed before about special right triangles is how even though you cut them down, its value will always stay consistent. 

2. Being able to derive these patterns myself aids in my learning because it helps me understand where everything comes from and how one side effects the other sides rather than just itself.