Problem 8 · 2001 AMC 8
Medium
Geometry & Measurement
spatial-reasoningproportion
To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. The kites look like the one in the diagram. For her small kite Genevieve draws the kite on a one-inch grid (shown below). For the large kite she triples both the height and width of the entire grid.
Genevieve puts bracing on her large kite in the form of a cross connecting opposite corners of the kite. How many inches of bracing material does she need?
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Answer: E — 39 inches.
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Hint 1 of 2
"A cross connecting opposite corners" is just the kite's two diagonals — you already know their grid lengths from before.
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Hint 2 of 2
Tripling the grid is a scale factor of 3. Lengths scale by 3 (not 9 — that's areas), so each diagonal triples.
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Approach: the cross is the two diagonals, scaled up ×3
- The bracing follows the two diagonals. On the small grid they're 6 and 7 units.
- Tripling every length multiplies each diagonal by 3: 6 → 18 and 7 → 21 inches.
- Total bracing = 18 + 21 = 39 inches. Key idea: under a scale factor k, lengths multiply by k while areas multiply by k² — so triple the grid means ×3 for bracing but ×9 for any area.
Another way — scale the small total at the end:
- On the small kite the two diagonals total 6 + 7 = 13 units.
- Tripling all lengths multiplies that total by 3: 13 × 3 = 39 inches.
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