Problem 12 · 1985 AJHSME
Hard
Geometry & Measurement
equal-perimetersquare-area
A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are 6.2 cm, 8.3 cm, and 9.5 cm. The area of the square is
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Answer: B — 36 cm².
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Hint 1 of 2
The phrase 'equal perimeters' is the only bridge between the triangle and the square. Find the triangle's perimeter first — that number is also the square's perimeter, even though you can't see the square.
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Hint 2 of 2
A square's four sides are all equal, so its side is (perimeter ÷ 4). Once you have the side, the area is side². Watch the order: perimeter → side → area, not perimeter → area directly.
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Approach: perimeter → side → area
- Add the triangle's three sides: 6.2 + 8.3 + 9.5 = 24 cm (the decimals were chosen to total a whole number). Equal perimeters means the square also has perimeter 24.
- A square splits its perimeter into 4 equal sides: side = 24 ⁄ 4 = 6 cm. Then area = 6² = 36 cm².
- Spot the trap: the answer is 6² = 36, not 6×… or 24 itself. Choices like 64 and 144 are squares of 8 and 12 — they're there to catch a wrong side length, so double-check you divided the perimeter by 4.
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