I/D# 2: Unit O Concepts 7-8: Deriving Patterns for the SRTs

Inquiry Activity Summary

1.) 30-60-90 SRT

We start with an equilateral triangle with side lengths 1 and split it in half in a way that bisects an angle and is perpendicular to the side opposite.

Each angle has a value of 60º (so bisecting them would get two 30º angles) and each side has a value of 1 (so making the 'cut' perpendicular to the side opposite would get you two 90º angles and side lengths of 1/2.)

Use the Pythagorean theorem (a^2 + b^2 = c^2) to find the missing leg length. 1/2 (let's say it's a) squared is 1/4, and 1 (let's say it's c) squared is just 1. When you subtract 1/4 from 1 to solve for b you get 3/4, the square root of which is radical 3 over 2.

To get rid of the fractions and make life easier for ourselves we multiply all the terms by 2. This results in the shortest leg equaling 1, the other leg equaling radical 3, and the hypotenuse equaling 2. We now set n equal to 1 (we choose n because it's a variable and we choose 1 because it's in all of the terms) so we can use it for number values other than 1, 2 and radical 3. Therefore, the pattern is n (shortest leg), 2n (hypotenuse- think of it as 2 times 1 and replace the 1 with n), and n radical 3 (other leg- again, think of it as 1 times radical 3 and replace the 1 with n.)

2.) 45-45-90 SRT

We start with a square with side lengths of 1 and split it in half diagonally.

This bisects the corner angles, which have a value of 90º each. 90/2 = 45º.

We know the two legs of the triangle each equal 1, so to find the hypotenuse we use the Pythagorean theorem (a^2 + b^2 = c^2). a and b represent the legs of the triangle, so we set them both equal to 1; 1 squared is just 1, so 1 + 1 = 2. c therefore equals radical 2.

We now set n equal to 1 (we choose n because it's a variable and we choose 1 because it's in all of the terms) so we can use it for number values other than 1 and radical 2. Therefore, the pattern is n (one leg), n (other leg), and n radical 2 (for the hypotenuse- think of it as 1 times radical 2 and replace the 1 with an n.)

Inquiry Activity Reflection

1.) “Something I never noticed before about special right triangles is…” that they were derived from an equilateral triangle and a square.
2.) “Being able to derive these patterns myself aids in my learning because…” I now know how to derive the side lengths myself instead of relying on my subpar memory.