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

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Problem

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. 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?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Small-kite vertices on a one-inch grid: $(3,0)$, $(0,5)$, $(3,7)$, $(6,5)$; Large kite: every length on the grid is tripled (scale factor $3$); Bracing $=$ the two diagonals of the large kite, joined into a cross; Answer choices: (A) $30$, (B) $32$, (C) $35$, (D) $38$, (E) $39$

Unknowns: Total length, in inches, of the bracing material for the large kite

Understand

Plan

Primary tool: #1 Draw a Diagram

Secondary: #9 Solve an Easier Related Problem

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.

Execute — Answer: E

#1 Draw a Diagram 5.G.A.1 Step 1

Use the grid to read the two diagonals of the small kite.
The vertical diagonal joins $(3,0)$ to $(3,7)$, so its length is the difference in $y$-coordinates.
The horizontal diagonal joins $(0,5)$ to $(6,5)$, so its length is the difference in $x$-coordinates.

$$\text{Vertical diagonal} = 7 - 0 = 7 \text{ in}, \quad \text{Horizontal diagonal} = 6 - 0 = 6 \text{ in}$$

💡 Both diagonals run along grid lines, so reading their length is just counting unit dots between the endpoints — exactly the Grade 5 coordinate-plane skill.

#9 Solve an Easier Related Problem 5.NF.B.5 Step 2

Scale up to the large kite.
Tripling the grid multiplies every length by $3$, so each diagonal of the large kite is $3$ times the corresponding small-kite diagonal.

$$\text{Large vertical} = 3 \times 7 = 21 \text{ in}, \quad \text{Large horizontal} = 3 \times 6 = 18 \text{ in}$$

💡 Solving the small (easier) kite first turns the large kite into a one-step scaling: multiply each known diagonal by the factor $3$.

#1 Draw a Diagram 4.OA.A.3 Step 3

The bracing is the two diagonals of the large kite, so the total length of bracing is the sum of those two diagonals.

$$21 + 18 = 39 \text{ in} \;\Rightarrow\; \textbf{(E)}$$

💡 A cross made of two diagonals uses one piece for each diagonal, so adding the two diagonal lengths gives the total material — a one-step Grade 4 word-problem sum.

[1] #1 5.G.A.1 Use the grid to read the two diagonals of the small kite. The vertical diagonal

[2] #9 5.NF.B.5 Scale up to the large kite. Tripling the grid multiplies every length by $3$, so

[3] #1 4.OA.A.3 The bracing is the two diagonals of the large kite, so the total length of braci

Review

Reasonableness: Check the scaling a different way. The whole small kite fits in a $6 \times 7$ bounding box, so tripling gives an $18 \times 21$ 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 $39 = 3 \times 13$ is, while distractors $32$, $35$, and $38$ are not. Both checks point to (E).

Alternative: Tool #7 (Identify Subproblems): treat "diagonals of the large kite" as one subproblem (find each diagonal: $21$ and $18$) and "total bracing length" as a second subproblem (add them: $39$). Solving the two pieces in order and combining gives the same $39$ inches, confirming (E).

CCSS standards used (min grade 5)

5.G.A.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system; identify points in the plane by ordered pairs (Reading the small-kite vertices $(3,0)$, $(0,5)$, $(3,7)$, $(6,5)$ off the one-inch grid and computing each diagonal as a difference of coordinates ($7$ and $6$).)
5.NF.B.5 Interpret multiplication as scaling (resizing) (Recognizing that tripling the grid multiplies every length by $3$, so the small-kite diagonals $7$ and $6$ become $21$ and $18$ on the large kite.)
4.OA.A.3 Solve multistep word problems posed with whole numbers using the four operations (Combining the two large-kite diagonal lengths into the total bracing length $21 + 18 = 39$ inches.)

⭐ 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).

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 5)

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