SOLUTION: if the length of the hypotenuse of a 30-60-90 triangle is 7 square root of 2 what's the area and perimeter. I have to find proofs for the problem for modern geometry class.

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Question 801311: if the length of the hypotenuse of a 30-60-90 triangle is 7 square root of 2 what's the area and perimeter. I have to find proofs for the problem for modern geometry class.
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Draw the triangle. This is half of an equilateral triangle. The hypotenuse of the 30-60-90 triangle is two times the short leg.

Let y=height
Let x=short leg
Already known, 7sqrt%282%29+=+h, hypotenuse, same as side of equilateral triangle.

As property of 30-60-90 triangle, h=2x.
According to pythagorean theorem, x%5E2%2By%5E2=h%5E2
We already know h, so we want to use the h relationship formula to make equation with only one unknown variable in the pythagorean theorem relationship.
x=h%2F2, so substitute:
%28h%2F2%29%5E2%2By%5E2=h%5E2
y%5E2=h%5E2-%28h%2F2%29%5E2
y%5E2=4h%5E2%2F4-h%5E2%2F4=%281%2F4%29%284h%5E2-h%5E2%29
y%5E2=%28h%5E2%2F4%29%283%29
y=%28h%2F2%29sqrt%283%29

Now go back and find x=h%2F2. You will find area as %281%2F2%29%28h%2F2%29%28%28h%2F2%29sqrt%283%29%29 which is simply %281%2F2%29%2Abase%2Aheight.