Problem 9 · 2015 Math Kangaroo
Medium
Geometry & Measurement
pythagorean-triplearea
The diameters of three semi-circles form the sides of a right-angled triangle. Their areas are X cm², Y cm² and Z cm² as pictured. Which of the following expressions is definitely correct?
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Answer: C — X + Y = Z
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Hint 1 of 2
A semicircle's area depends on the square of its diameter.
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Hint 2 of 2
The diameters are the triangle's sides, which satisfy the Pythagorean relation.
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Approach: semicircle areas track the squares of the sides
- Each semicircle area is πd²/8, proportional to the square of its diameter (a side of the right triangle).
- For a right triangle, leg² + leg² = hypotenuse², so the two smaller semicircle areas add to the largest.
- With Z on the hypotenuse, X + Y = Z (C).
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