Problem 12 · 1994 AJHSME
Hard
Geometry & Measurement
dissectionarea-comparison
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Answer: A — All three are equal.
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Hint 1 of 2
Resist computing three messy areas. Since the cuts all hit the midpoints, the shaded bits in each square are made of the same little half-of-a-quarter triangles — slide them around in your head.
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Hint 2 of 2
Find what fraction of ONE square is shaded, then check the other two land on the same fraction. (Picture square II: it's clearly one quarter shaded — that's your target to match.)
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Approach: cut and rearrange to compare
- Square II is the easy anchor: one of the four equal quarters is shaded, so it's exactly 1/4 of the square.
- Square I: the two shaded triangles each sit in a quarter of the grid and fill half of it (the midpoint cut halves them), so together they cover 1/4. Square III: the shaded diamond pieces likewise reassemble — slide the corner triangles inward — to cover 1/4.
- All three are all equal (each is 1/4 of the square).
- Big idea: when every cut goes through a midpoint, shapes are built from identical building-block triangles. Rearranging those blocks (instead of computing each odd shape) is the fast, error-proof way to compare areas — you'll reuse this 'cut-and-slide' trick constantly in geometry.
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