Problem 21 · 2012 Math Kangaroo
Stretch
Geometry & Measurement
perimeterarea-decomposition
Both the figures on the right were made out of the same 5 pieces. The rectangle has dimensions 5 cm × 10 cm. The other pieces are quarter circles with 2 different sized radii. What is the difference between the perimeters of the two figures?
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Answer: E — 20 cm
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Hint 1 of 2
Both shapes use the very same five pieces, so the curved (arc) parts of their outlines are identical.
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Hint 2 of 2
Only the straight rectangle edges show up differently, so compare just those.
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Approach: cancel the identical arcs, compare straight edges
- Both figures are built from the very same five pieces, so the curved quarter-circle arcs contribute the same total length to each outline and simply cancel when we take the difference.
- The only thing that can differ is how much of the rectangle's straight edges shows on the outside; in one figure a pair of the rectangle's 10 cm sides lies on the boundary while in the other they are tucked inside.
- That swap of two 10 cm straight edges gives a perimeter difference of 2 x 10 = 20 cm (choice E).
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