A circle with four chords. Two chord lengths are given. What is the circle area?

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The circle area is 5π.

Solution by ÜmitYILDIRIM.

First, construct the angles α and β subtended at the circumference by the given chords. It is quite easy to see from the triangle interior angle sum that α+β=45°.

Now rotate the short chord all the way along the circle until it sits next to the longer chord. Their total angle subtended at the circumference is still 45°. Because they form a cyclic quadrilateral, the angle between the chords must be 135°.

From the Inscribed angle theorem it follows that the angle subtended at the centre is 90°. The combined chord HL therefore has length r√2. Applying the cosine rule in triangle HLF gives 2r2=4+2-√2cos(135)=10. The circle area follows.


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