Problem 3 · 1990 AJHSME
Hard
Geometry & Measurement
area-fractionrearrangement
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Answer: E — 1/2.
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Hint 1 of 2
Trying to measure each shaded chunk separately is slow. Instead, look at the long diagonal running corner to corner — what does it do to the whole square?
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Hint 2 of 2
Use symmetry, not arithmetic: the diagonal splits the square into two equal triangles, and the figure is built so the shading on one side of the diagonal exactly mirrors the unshaded part on the other.
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Approach: let the diagonal do the work (symmetry over computing pieces)
- Don't compute the little triangles and rectangle one by one. Notice the main diagonal cuts the square into two congruent triangles — each is exactly half the square.
- Along that diagonal the picture is balanced: every shaded piece on one side is matched by a same-size unshaded piece on the other. Slide the shaded pieces together and they fill exactly one of the two half-triangles.
- So the shaded part is one half of the square: 1/2.
- *Why this transfers:* when a figure has a line of symmetry (a diagonal, a center line), look for shaded/unshaded pieces that pair up across it — the fraction is often a clean 1/2 with no measuring at all.
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