Problem 24 · 2025 Math Kangaroo
Hard
Geometry & Measurement
area-decompositionarea
The square ABCD contains two shaded rectangles (see diagram). The dimensions are as shown and the area of the overlapping region is 18 cm². What is the perimeter of the square ABCD?
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Answer: C — 36 cm
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Hint 1 of 2
Let the square's side be \(s\); the top-left rectangle is \(7\times5\) and the bottom rectangle is \(8\times7\).
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Hint 2 of 2
Find the overlap's width and height in terms of \(s\) by seeing how far the two rectangles reach into each other, then set that product equal to 18.
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Approach: overlap area gives an equation for the side
- The top-left rectangle reaches 7 cm in from the left and 5 cm down from the top; the bottom rectangle reaches 8 cm in from the right and 7 cm up from the bottom.
- Their overlap is therefore \((7+8-s)\) wide by \((5+7-s)\) tall, i.e. \((15-s)(12-s)\), and this equals 18.
- Solving \((15-s)(12-s)=18\) gives \(s=9\) (the other root is too big to fit), so the perimeter is \(4\times 9=\) 36 cm, answer C.
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