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

**INQUIRY ACTIVITY SUMMARY**

**1.**** 30-60-90 triangle:** We are given an equilateral triangle with a side length of 1. I drew a vertical line straight down the center of the equilateral triangle to get 2 30-60-90 triangles.
If I look at one of the triangles, I know that the hypotenuse (the side across from the 90* angle) is 1 and the shortest side is 1/2 because we cut the side in half. To find the height , opposite of the 60* angle, I used the Pythagorean theorem, a^2+b^2=c^2.
I plugged in 1/2 for a and 1 for c. Squaring 1/2 gave me 1/4 and squaring 1 is 1. I wanted to solve for b^2 so i subtracted 1/4 from both sides, giving me 3/4. Next, I took the square root of both sides, leaving me with b=rad3/2. To translate this into a normal pattern( without fractions, I multiplied all the side lengths by 2. This gave me a hypotenuse of 2, the shortest leg was 1 and the height was rad3.
Lastly, I added n to all of the side lengths. This is so that this concept can be applied to all 30-60-90 triangles. By adding n, we emphasize the relationship between the sides, so that we can still solve for sides with different values (not just one) by applying the ratio.
**2. 45-45-90 triangle:** When given a square with a side length of 1, we can derive the patterns for this special right triangle. First, I drew a diagonal line connecting 2 opposite angles to form 2 45-45-90 degree triangles.

Knowing the length of 2 of the sides, which is 1, I used the Pythagorean Theorem (a^2+b^2=c^2) to solve for the hypotenuse. I plugged 1 into a and 1 into b. First I squared them giving me 1+1=c^2 -> 2=c^2. Next, I took the square root of both sides to give me the hypotenuse of rad2.
Now that I have the length for all 3 sides, I added n to each of the lengths so that it can apply to all 45-45-90 triangles when the side length is not 1. Through using this ratio, the relationship of the 45-45-90 degree angles will remain constant with the measure of the sides.
**INQUIRY ACTIVITY REFLECTION**
1. **Something I never noticed before about special right triangles is** how the ratios came to be. I never realized that the ratios came from the Pythagorean theorem!
2.** Being able to derive these triangles myself aids in my learning because** I now conceptually understand the roots of the rules of special right triangles. Through this activity, I am not simply memorizing information, because I now understand where these values came from. Even if I forget the ratios, I will be able to derive these patterns.

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