Problem 11 · 2022 Math Kangaroo
Hard
Geometry & Measurement
pythagorean-triplearea
Five squares and two right-angled triangles are placed as shown in the diagram. The numbers 3, 8 and 22 in the squares state the size of the area in m². How big is the area (in m²) of the square with the question mark?
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Answer: D — 17
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Hint 1 of 2
Each right triangle ties three squares together through the Pythagorean relation on its sides.
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Hint 2 of 2
Chain the area relations across the two triangles to reach the '?' square.
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Approach: use that each right triangle makes leg-squares add to the hypotenuse-square
- For a right triangle, the square on the hypotenuse has area equal to the sum of the squares on the two legs.
- The two triangles share a side, so the chain of squares gives \(? + 8 = 22 + 3\): the unknown plus 8 equals the other two squares combined.
- Hence \(? = 22 + 3 - 8 = 17\), answer D.
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