SOLUTION: a circle is inscribed in a square which is circumscribed by another circle. If the diagonal of the square is 2x, find the ratio of the area of the large circle to the area of the s

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Question 879092: a circle is inscribed in a square which is circumscribed by another circle. If the diagonal of the square is 2x, find the ratio of the area of the large circle to the area of the small circle?
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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a circle is inscribed in a square which is circumscribed by another circle.
If the diagonal of the square is 2x, find the ratio of the area of the large circle to the area of the small circle?
:
Draw this out, observe that the diameter of the large circle is equal to the
diagonal of the square, 2x, therefore the x = the radius of the large circle
:
The side of the square is equal to the diameter of the small circle. Find the
side (s) of the square using the diagonal of the square; pythag.
s^2 + s^2 = (2x)^2
2s^2 = 4x^2
divide both sides by 2
s^2 = 2x^2
s = sqrt%282x%5E2%29
s = x%2Asqrt%282%29, the diameter of the small circle
therefore
%28x%2Asqrt%282%29%29%2F2 = the radius of the small circle
:
" find the ratio of the area of the large circle to the area of the small circle?"
Cancel pi,
%28pi%2Ax%5E2%29%2F%28pi%28%28x%2Asqrt%282%29%29%2F2%29%5E2%29 = x%5E2%2F%28%282x%5E2%29%2F4%29%29 = x%5E2%2F%28x%5E2%2F2%29 = x%5E2+%2A+%282%2Fx%5E2%29 = 2
Area of large circle to small is 2:1