A right triangle has legs 5 cm and 12 cm, so its hypotenuse is 13 cm. A semicircle is drawn outwards on each leg as diameter, and a semicircle is drawn on the hypotenuse as diameter on the same side as the triangle, so that it covers the triangle. Two crescent-shaped regions (lunes) are left between the leg semicircles and the hypotenuse semicircle. Find their combined area in cm².

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Accepted Answer

30. Let S₁, S₂ be the semicircles on the legs and S₃ the semicircle on the hypotenuse. Since 5² + 12² = 13², we get S₁ + S₂ = S₃ (semicircle areas are proportional to the squares of their diameters). The two lunes together = (S₁ + S₂ + triangle) − S₃ = triangle = ½ × 5 × 12 = 30 cm².

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