Problem 17 · 2020 Math Kangaroo
Medium
Geometry & Measurement
Algebra & Patterns
difference-of-squaresarea-decomposition
A square is formed by four identical rectangles and a central square, as in the figure. The area of the large square is 81 cm², and the square formed by the diagonals of these rectangles has area 64 cm². What is the area of the central square?
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Answer: D — 47 cm²
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Hint 1 of 2
Call the rectangle sides a and b; one big side is a + b = 9, and the diagonal is √(a²+b²) = 8.
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Hint 2 of 2
The central square has side a − b, so its area is a² + b² − 2ab.
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Approach: combine the sum and the diagonal
- The outer square has side 9 (area 81), so a + b = 9 and (a+b)² = 81.
- The diagonals' square has side 8, so a² + b² = 64.
- From 81 = 64 + 2ab we get 2ab = 17, and the central square area is (a−b)² = a²+b² − 2ab = 64 − 17 = 47.
- The central square is 47 cm², choice D.
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