I/D #2: Unit O Concept 7 & 8- Special Right Triangles

Inquiry Activity Summary 

1.  We get a 30-60-90 triangle by cutting an equilateral triangle in half. We start off by using the Pythagorean theorem, which is a^2 + b^2 = c^2. By using this we know that our a is going to be 1 and our b is still unknown. Our c is going to be 2 as seen in the picture above.  Since we know our a and c we can plug those numbers into the Pythagorean theorem for our b. As seen in the picture above after plugging in those two values we find our b which is rad3.

As seen in this picture above we found a pattern with the 30-60-90 triangles. We added a variable at the end of each value side because we noticed that it can work for any number possible. We just start off with n to get a, then times it by rad3 to get b, and by 2 to get c. We multiplied each variable by 2 to get rid of the fractions so it can be simpler as well. 

2. We get a 45-45-90 triangle by drawing a slanted line through a square. We start off using the Pythagorean theorem, which is a^2 + b^2 = c^2. Since it is a square all sides are the same. For this example our sides are going to be 1. So we must plug in 1 into a and b to find c. As seen in the picture above, the work shows that our c is rad2.

As seen in this other picture above, we found a pattern with the 45-45-90 triangles. We added a variable, which is n to each value because the sides will not always be 1. This can work for any number because it is derived from the square. We put “n” in the pattern because it can be any number if it follows the 45-45-90 pattern. 

Inquiry Activity Reflection 

1. Something I never noticed before about special right triangles is where these patterns for them actually came from. After doing this activity I found out the origins of the patterns, which makes sense to me now.  I never knew the 45-45-90 came from a square and the 30-60-90 came from an equilateral triangle.

2. Being able to derive these patterns myself aids to my learning because it kind of ties everything together now. It even ties in together to unit circle because of the vales of the triangles. It also helps me remember the patterns now thanks to the Pythagorean theorem.