Lunes, Version 1.1

Geometry Level 2

ABC (in green) is a right triangle. A circumcircle of ABC and half circles drawn on its legs generate two lunes (in blue). Which colored area is larger?

Green and blue areas are the same size The relationship of the areas depends on the shape of triangle ABC Green area is larger than blue area Blue area is larger than green area

Marta Reece by Marta Reece , 69, USA

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1 solution

Marta Reece
Jan 5, 2018

If the legs of the triangle are a a and b b , then the area of the lunes is the two semicircles, π 2 a 2 + π 2 b 2 \frac{\pi}2a^2+\frac{\pi}2b^2 , plus the triangle a b 2 \frac{ab}2 , minus the white semicircle π 2 c 2 \frac{\pi}2c^2

But π 2 a 2 + π 2 b 2 + a b 2 − π 2 c 2 = π 2 ( a 2 + b 2 − c 2 ) + a b 2 = π 2 × 0 + a b 2 = a b 2 \frac{\pi}2a^2+\frac{\pi}2b^2+\frac{ab}2-\frac{\pi}2c^2=\frac{\pi}2(a^2+b^2-c^2)+\frac{ab}2=\frac{\pi}2\times0+\frac{ab}2=\frac{ab}2

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