Q: The polygon circumscribes the circle. Find the perimeter.

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What do you know about 2 tangents to a circle intersecting at a point? –  Mike Dec 16 '12 at 21:25
What are the measurements of exactly. What do you want the perimeter of? –  Lucas Dec 16 '12 at 21:25
I'm looking for the perimeter of the square, and I don't ever remember seeing this problem. –  Nathan Abbott Dec 16 '12 at 21:26

It is a fact that any two intersecting tangents are equal in length.

Thus, the perimeter is $2(9+7+5+8)=58$

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It's a basic theorem in euclidean geometry: a quadrilateral circumscribes a circle iff the sum of the lengths of both pairs of opposite sides is the same. This follows at once from the fact that Amr reminds in his answers: both tangents to a circle from an exterior point to the circle have the same length.