I/D #2: Unit O - Deriving the SRTs

INQUIRY ACTIVITY SUMMARY

30-60-90: The equilateral triangle has all 60* angles and is labeled to have the length of 1 on all sides. You are looking for a 30-60-90 triangle so you cut the triangle in half vertically. You will have a 30* angle, a 60* angle and a 90* angle.

You see that because you had cut the triangle in half, the side across from 30* is 1/2 and the side across from 90* is 1.

You then half to find the hypotenuse, so you use the Pythagorean Theorem to solve for the missing side. 

The side across from 60* ends up being radical 3 over 2. To get rid of the fractions, you multiply everything by 2. You end up with 30* being 1, 90* being 2 and 60* being radical 3. The "n" that is used is to express that any number can be put in place of "n". Meaning that the 30-60-90 triangle can have any lengths, bigger or smaller, than the triangle used in this example and that it will still have the same pattern as long as it is a 30-60-90 triangle. Through this pattern, when there is a 30-60-90 triangle, the side across 30* will be "n", while 90* will be 2 times "n" and 60* will be "n" times radical 3.

45-45-90: The equilateral square has sides all the length of 1. To change this into a 45-45-90 triangle, you have to cut it diagonally to get the 45-45 degree angles.

You see that because you had cut the square diagonally, both the side across from 45* and 45* are still 1. So now you have to use Pythagorean Theorem to solve for the missing side, 90*.

The missing side across 90* ends up being radical 2. The "n" is used to express that there are many numbers that can be used in "n" place. The lengths of the triangle can be longer or shorter than 1 and will still maintain the same pattern as long as it is a 45-45-90 triangle. Through this pattern, when there is a 45-45-90 triangle, the sides across from both 45* and 45* angles will be "n" while the side across form 90* angle will be n times radical 2.

INQUIRY ACTIVITY REFLECTION

Something I never noticed before about special right triangles is how the pattern is determined. I faintly remember how the Pythagorean Theorem was derived, but looking at how using the Pythagorean Theorem can help derive how the pattern is created is interesting.
Being able to derive these patterns myself aids in my learning because if for some reason I forget the pattern, I can draw the triangle/square and derive the pattern out and use that instead of guessing the pattern.