AMC 8 · 2001 · #8

Grade 5 geometry-2drate-ratio
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
similar-figurescoordinate-geometrymulti-digit-arithmetic identify-subproblemscoordinate-geometry ↑ Prerequisites: similar-figurescoordinate-geometry
📏 Medium solution 💡 3 insights 📊 Diagram
📘 View easy version →
Problem
Genevieve draws a small kite on a one-inch dot grid with vertices at (3,0), (0,5), (3,7), and (6,5). For the large kite she triples every length on the grid. She then braces the large kite with a cross along its two diagonals (each diagonal connects opposite corners). How many inches of bracing material does she need?

Pick an answer.

(A)
30
(B)
32
(C)
35
(D)
38
(E)
39

AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The figure is on a one-inch grid, so Tool #1 (Draw a Diagram) is built in — read each diagonal straight off the grid by counting units between opposite vertices. Tool #9 (Solve an Easier Related Problem) handles the scaling: get the answer for the easy small kite first, then multiply by 3 for the large kite. Together these two tools avoid any need for algebra or distance formulas.

1STEP 1

Read the small kite's two diagonals off the grid by counting units: the vertical is 7 in and the horizontal is 6 in.

Vertical diagonal = 7 - 0 = 7 in, Horizontal diagonal = 6 - 0 = 6 in
2STEP 2

Tripling the grid multiplies every length by 3, so the diagonals become 21 in and 18 in.

Large vertical = 3 × 7 = 21 in, Large horizontal = 3 × 6 = 18 in
3STEP 3

The cross is the two large diagonals, so add them: 21 + 18 = 39 in of bracing.

21 + 18 = 39 in → (E)
Answer
39
Check the scaling a different way. The whole small kite fits in a 6 × 7 bounding box, so tripling gives an 18 × 21 bounding box for the large kite. The horizontal diagonal spans the full width (18), and the vertical diagonal spans the full height (21); their sum is 18 + 21 = 39. Also, the answer should be a multiple of 3 since every length on the large kite is 3 times an integer — and 39 = 3 × 13 is, while distractors 32, 35, and 38 are not. Both checks point to (E).
💡Key takeaway

On a grid, a diagonal that runs straight across rows or columns is just a counting job — 7 up, 6 across. Triple the grid and you triple each length (21 and 18), then add the two diagonals for the cross-shaped brace: 21 + 18 = 39 inches, choice (E).

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