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Area of a Rhombus

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A rhombus, ABCD, has sides of length 10. A circle with center A passes through C (the opposite vertex.) Likewise, a circle with center B passes through D. If the two circles are tangent to each other, what is the area of the rhombus?

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The area is 75

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The area is 75

Care to share your solution.

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Let T be the point of tangency of two circles. The key to the solution is to notice that the points A, B and T are collinear. AT is the radius of the large circle and is perpendicular to the tangency line. BT is the radius of the small circle and is also perpendicular to the tangency line. We know that AB=10, AT=AC and BT=BD, so from this we can conclude that AC=10+BD.

Let O be the center of the rhombus. Then BO=BD/2 and AO=AC/2=BD/2+5=BO+5. The area of the rhombus is 2*AO*BO.

From AO2+BO2=AB2 follows (BO+5)2+BO2=100. Expanding and simplifying we get 2*BO2+10*BO=75 or 2*BO*(BO+5)=75. The left side is actually the area of the rhombus.

QED

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