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Rectangular Circles

Problem

Given rectangle OABC, three concentric circles OA, OB, and OC are drawn.


Prove that the area of the annulus generated by the concentric circles OB and OC is always equal to the area of the inner circle OA.


Solution

Using the Pythagorean theorem it is clear that (OB)2 = (OA)2 + (OC)2, or (OA)2 = (OB)2 minus (OC)2.

therefore πtimes(OA)2 = πtimes(OB)2 minus πtimes(OC)2

That is, we haven shown that the area of circle OA is equal to the area of the annulus as required.

If the area of the central annulus generated by circles OC and OA is also equal to the area of the inner circle OA, what can be deduced about the ratio of the sides of the rectangle?

Problem ID: 253 (12 Dec 2005)     Difficulty: 2 Star

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