Problem 15 · 2000 AMC 8
Hard
Geometry & Measurement
perimeter
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Answer: C — 15.
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Hint 1 of 2
'Midpoint' is the key word: building the next triangle on a midpoint makes its side exactly half. So the sides go 4, 2, 1 — each triangle half the last.
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Hint 2 of 2
Perimeter means the OUTSIDE boundary only. Don't add all three full triangles — where one triangle sits on another, that shared edge is interior and gets skipped. Just walk the outline once.
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Approach: walk the outer boundary, skipping shared edges
- Each new triangle is built on a midpoint, so its side is half the previous: 4, then 2, then 1.
- Trace the outline A→B→C→D→E→F→G→A and record only edges on the outside. You get 4, 4 (big triangle's two outer sides), 2, 2 (middle), 1, 1, 1 (small), summing to 15.
- Sanity check / the principle: if you'd naively added all three perimeters (12 + 6 + 3 = 21, choice E — the trap!), you'd double-count the edges where triangles meet. Perimeter is always the outer boundary, so interior shared segments never count.
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