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

Inquiry Activity Summary

45-45-90 Triangle

First, I cut the square diagonally. I labeled all the angles 90 and the ones who got cut in half 45. 

They all add up to 360 like a square should.Each side length is 1 so we label each side one. ( Step one 

Picture) The diagonal side we don't know yet but since we know two sides are one as stated in the 

directions, we can figure out the diagonal side by using the pythagorean theorem, a^2 + b^2 = c^2. so 

its 1+1=c^2 so then c=sqaureroot 2. (Step two picture) Why "n"? In the sss packet we were given a 45-

45-90 triangle with side lengths of n,n, and n rad 2. Why is that? N is being used as a variable so that it 

can work in all problems with different types of numbers. In this example to get n,n, and nrad2, i just 

multiplied each by n.(Step 3 picture)

30-60-90 Triangle

First , I cut the equilateral triangle straight down the middle. Then I labeled each side and angle 

according to what i was given. Since each side was one and the bottom got cut in half, it is now 1/2. 

We now know 2 sides. One side is 1/2 and one side is 1. We can use the Pythagorean Theorem to 

figure it out but this time we know a and c and are looking for b. (Math work on picture)Finally since 

we now know b is  rad 3/2 and a is 1/2 and c is 1 we need to derive them so that they corresponds to 

all numbers. So what i did was i multiplied each one by 2n. A which was 1/2 became n, b which was 1 

became 2n, and c which was rad 3/2 became n rad 3. (Math work shown on picture). The reason why 

we need to have n is because not all problems will have 1 as the numbers, it has to be able to work for 

all numbers so n acts as the variable.

Inquiry Activity Reflection

Something I never notcied before about special right traingles are that they actaully originate from something. I didn't know that the 45-45-90 came from a sqaure or that the 30-60-90 came from another traingle. I thought that these triangles were just traingles.

Being able to derive these patterns myself aids in my learning beacuse once I know how to get the varaibles that go with these traingles, solving for the sides will be easier now that I know how to get them in the first place. Solving these traingles is no longer just memorization but application of math.