SOLUTION: A square and an equilateral have equal perimeter. If the area of the triangle is 9 sqrt(3) cm squared, find the exact length of the diagonal of the square.

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Question 1121142: A square and an equilateral have equal perimeter. If the area of the triangle is 9 sqrt(3) cm squared, find the exact length of the diagonal of the square.
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
This can be only a way to get started, if you want:
Let 2x be the side length of the equilateral triangle.
Let y be the altitude or height of the equilateral triangle.

system%28%281%2F2%29xy=9sqrt%283%29%2Cy%5E2%2Bx%5E2=4x%5E2%29

system%28xy=9%2Asqrt%283%29%2Cy=x%2Asqrt%283%29%29

highlight_green%28x=3%29 and therefore side of equilateral triangle highlight_green%282x=6%29.

You continue from here...
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Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The area of an equilateral triangle with side length s is %28s%5E2%29%2A%28sqrt%283%29%2F4%29.

Give that the area of the triangle is 9*sqrt(3), the side length is 6:
%28s%5E2%29%2A%28sqrt%283%29%2F4%29+=+9%2Asqrt%283%29 --> s%5E2%2F4+=+9 --> s = 6.

Now find the perimeter of the triangle, which is the same as the perimeter of the square; so you can find the length of a side of the square and then find the length of the diagonal.