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A rectangle WXYZ and a circle are given. The circle is tangential to WX and WZ. Vertex Y lies on the circle. The circle crosses YZ in point A. Determine the area of the trapezoid WXAZ if WX=9 and WZ=8.

So I made a sketch: Sketch

But I can't stumble upon any idea. I thought about cutting the trapezoid somehow and applying Pythagorean theorem but it doesn't seem to work. Also finding the radius of the circle doesn't look easy.. How can I do this?

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3 Answers

up vote 1 down vote accepted

Hint: You want to find the radius of the circle. Let O be the center of the circle.

From O, let the line perpendicular to WX and ZY intersect WX and ZY at A and B.
From O, let the line perpendicular to WZ and XY intersect WZ and XY at D and C.

Observations:

WA = WD = XC = ZB = R. (R is the radius of the circle)

In the rectangle OCYB, you know OY is the radius of the circle, OC is 9-R, OB is 8-R, and you can use Pythagorean Theorem to solve the quadratic equation to find R. (which has 2 positive roots, but one is not good). The good one is 5 as wxu pointed out.

After you know R, then you need to find ZA to calculate the area. One way to do it is to let the line from A perpendicular to DO intersect DO at E. ZA = DE. You know DO = R, you need to find OE. you can apply Pythagorean Theorem again to triangle AEO to find OE.

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Great answer, I get it! Thank you very much :) –  Straightfw May 1 '12 at 17:33
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Hint: compare the areas of the trapezoid, triangle, and rectangle. You can also spot two chords made by XA and XY.

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Hint: It suffices to figure out the radius of the cirle ($r=5$). Let point $O$ be the center of the circle. Let point $B$ be the center of the line $AY$. Consider the triangle $OAB$.

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