Problem 4 · 2008 AMC 8
Easy
Geometry & Measurement
area-decomposition
In the figure, the outer equilateral triangle has area 16, the inner equilateral triangle has area 1, and the three trapezoids are congruent. What is the area of one of the trapezoids?
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Answer: C — 5.
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Hint 1 of 2
The three trapezoids are exactly what's left when you cut the little triangle out of the big one — you don't need any side lengths.
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Hint 2 of 2
"Congruent" is the key word: equal pieces means just divide the leftover evenly.
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Approach: subtract the hole, split the rest evenly
- The big triangle is the small triangle plus the three trapezoids, so the trapezoids together cover 16 − 1 = 15. No measuring needed — the area you can't see is simply "total minus the hole."
- They're congruent (identical), so each is 15 ÷ 3 = 5.
- Sanity check: the choices run 3–7, and 15 split three ways must be 5 — only one choice fits.
Another way — scale-factor intuition:
- Area 16 vs area 1 means the big triangle is 4× the small one in side length (since area scales as the square, √16 = 4).
- The whole figure is 16 small-triangle-areas worth; the center hole is 1, leaving 15 for the three trapezoids, so each is 5.
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