Problem 21 · 2023 Math Kangaroo
Stretch
Geometry & Measurement
square-areaarea-decompositionarea-fraction
The diagram shows a grey rectangle that lies within a bigger rectangle, touching its sides. Two corner points of the grey rectangle are the midpoints of the shorter sides of the bigger rectangle. The grey rectangle is made up of three squares that each have an area of 25 cm². How big is the area of the bigger rectangle, in cm²?
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Answer: D — 150
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Hint 1 of 2
Each small square has area 25, so its side is 5 and the grey rectangle is 15 by 5, area 75.
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Hint 2 of 2
A rectangle whose two opposite corners are midpoints of the big rectangle's sides covers exactly half of it.
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Approach: grey is an inscribed rectangle covering half the big rectangle's area
- Each square has side √25 = 5, so the grey rectangle is 15 by 5 with area 75 cm².
- Two opposite grey corners are midpoints of the big rectangle's short sides, and the other two lie on its long sides.
- Such an inscribed rectangle always encloses exactly half of the outer rectangle, so the big area is 2 × 75 = 150 cm².
- So the answer is 150 (D).
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