Problem 1 · 2025 AMC 8
Easy
Geometry & Measurement
symmetryarea-fraction
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Answer: B — 50%.
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Hint 1 of 2
You probably don't need to measure anything. Could the shaded and unshaded parts be hiding the same total area?
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Hint 2 of 2
Look for symmetry that swaps shaded with white: pair every shaded triangle with a matching white triangle the same size. If they pair up perfectly, the answer is forced.
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Approach: pair each shaded triangle with a white twin (symmetry)
- Don't measure anything — look for a partner. Each shaded triangle of the star has a congruent white triangle right beside it; shaded and white tile the grid in matched pairs.
- Matched pairs means shaded = white exactly, so the star covers half the 4 × 4 grid: 8 of 16 squares = 50%.
- Why this transfers: whenever a figure has a symmetry that swaps shaded with unshaded (a flip or a quarter-turn), the shaded fraction is forced to 1/2 — you never have to compute the area.
Another way — shrink to one quarter (MAA):
- The four-fold symmetry means the shaded fraction of the whole equals the shaded fraction of just the top-left 2 × 2 quarter.
- In that quarter, sliding one little triangle over makes it obvious the shaded part is exactly half — so the whole is 50% too.
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