# Geometry

A circle is circumscribed by a pentagon. The sides are all tangent to the circle. (It's a bit hard to explain without a picture, so I'll do my best.)

The pentagon is QSUWY. However, there are points in between all five sides. In SQ lies point R, SU point T, UW point V, WY point X, and QY with point Z.

It is given that QZ=9, YX=7, XW=12, UW=15, and SU=16. Find the perimeter of the pentagon. In all, I know the perimeter so far is 59, and the problem is I'm not sure how to solve for SQ and ZY. Help please? It's much appreciated.

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1. Even though you did not state it, I will assume that the points R,T,V... are the points of contact of the sides with the circle.

One of the basic properties of tangents and circles is that
If tangents are drawn to a circle from an external point P, and those tangents touch at A and B, then PA = PB

You have that situation 5 times in your figure.
so QZ = QR, YZ= YX, etc.

work your way all around, you will be able to find each segment of the pentagon.
I got 88, but check my arithmetic, I am terrible in arithmetic.

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posted by Reiny

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