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

Inquiry Activities Summary

1.) 30-60-90 triangle

First of all, to derive this triangle, I should start with an equilateral triangle. Which means they have all the sides equal to each other, in this case all the sides equal to 1. Not only it is equilateral but it is also equiangular, meaning the angle are the same measure as well. So for an equilateral and equiangular triangle they must add up to 180°.

First we start with a equilateral triangle, which where are the sides equal to 1.  Then split the triangle in half, which will give me a 30-60-90 triangle. Which makes the base of the triangle equal to 1/2. So, what we have already know is "a" which equal to 1/2 and is the adjacent. I also know "c," which is 1, which is the hypotenuse. In order to find "b," I must use the Pythagorean Theorem. The Pythagorean Theorem is a^2 + b^2 = c^2. Then, I plug a, and c into the Pythagorean Theorem, a = 1/2, b = ?, and c = 1, and I got 1/4 + b^2 = 1. Then subtract 1/4 and got 3/4, and then take the square root of 3/4 and got radical 3/2.

Once, I got b, which equal to radical 3/2, then I multiply 1/2, radical 3/2, and 1 by 2 to get rid of fractions. So, I would have the hypotenuse equal to 2, the opposite equal to radical 3, and the adjacent equal to 1. Next, I take "n" multiply to 1, 2, and radical 3 because "n,"shows the relationships between the three sides of the triangle.  That will give me a the derive 30-60-90 triangle.

2.) 45-45-90 Triangle

In order to derive this square into a 45-45-90, we start off with a square with all equal sides. That means the square adds up to 360° In this case which also means that the sides equal to 1. Then split the square diagonally, so it will make it a 45-45-90 triangle. So the triangle have to add up to 180°. 

Based on the triangle, I already know what "a" and "b" equals, we just need to find "c." 

So, I know that a = 1 and b = 1, to find "c," we also use the Pythagorean Theorem just like I did to find  the 30-60-90 triangle. The Pythagorean Theorem is a^2 + b^2 = c^2. So, I plug it in the formula, 1^2 + 1^2 = c^2. Then, I got 2 equal c^2, so I have to take the square root and got radical 2 for c. Then I multiply n to 1, 1, and radical 2 and got n, n, and n radical 2. That will give me a derive 45-45-90 triangle.

Inquiry Activity Reflection:

1.) "Something I never notice before about special right triangle is..."the formula was found based and it is based on the Unit Circle.

2.) "Being able to derive these patterns myself aids in my learning because..."if I happen to forget how to solve for a special right triangle I can refer back to this.