45-45-90 Correction Text

Because the 45-45-90 is half a square, the leg lengths are identical.
For example, if the length of one leg of a 45-45-90 is 4, then the other leg is length 4 too.

DRAW A NEAT PICTURE OF A SQUARE BISECTED BY A DIAGONAL  

The length of the hypotenuse is found by using the pythagorean theorem. For example, 
if a leg is length 4 then:

1) a^2+b^2=c^2   
2) 4^2+4^2=c^2
3) 16+16=c^2
4) 32=c^2
5) sqrt(32)=sqrt(c^2)
6) 32= factor tree
7) sqrt(2^2*2^2*2)=sqrt(c^2)
8) 4sqrt(2) = c

Therefore the length of the hypotenuse in a 45-45-90 is the square root of 2 times the length of the leg.

Do this 3 times, using relatively prime numbers for the length of the leg. i.e., no multiples.

30-60-90 Correction Text

Because the 30-60-90 is formed by splitting an equilateral triangle in half, the length of the short leg of a 30-60-90
is half the length of the hypotenuse.

DRAW A NEAT PICTURE OF An EQUILATERAL CUT IN HALF 

For example, if the hypotenuse   of a 30-60-90 triangle is 8 , then the short leg of a 30-60-90 triangle is 4.

The length of the long leg of a 30-60-90 found by using the pythagorean theorem. For example if the short leg
is length 4 then:

1) a^2+b^2 = c^2
2) 4^2+b^2 = 8^2
3) 16+b^2=64
4) -16 = -16    / subtract 16 from both sides
5) b^2 = 48
6) sqrt(b^2) = sqrt(48)
7) factor tree of 48
8) sqrt(b^2) = sqrt(2^2*2^2*3)
9) b= 4sqrt(3)

Therefore the length of the long leg of a 30-60-90 is the square root of 3 times the length of the short leg.