AMC 8 · 2001 · #7

Grade 6 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Axis-Aligned Figure Area from Coordinates

coordinate-geometryarea-trianglesarea-rectanglesspatial-visualization coordinate-geometryidentify-subproblems ↑ Prerequisites: coordinate-geometryarea-triangles

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A kite is drawn on a one-inch grid. Reading the diagram, the four corners sit at (3,0), (0,5), (3,7), and (6,5). Find the area of this kite in square inches.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

How to solve

Strategy Draw a Diagram

The whole problem lives on the grid, so Tool #1 (Draw a Diagram) is the way in: place the kite on coordinates, read every length straight off the dots, and the horizontal segment from (0,5) to (6,5) jumps out as a natural cut. That cut splits the kite into a top triangle and a bottom triangle, which is Tool #7 (Identify Subproblems): one hard shape becomes two easy ones with a known base × height ÷ 2 formula.

1STEP 1

Put the bottom-left dot at (0,0) and read the four corners; L=(0,5) and R=(6,5) share height y=5, so segment LR is horizontal.

B=(3,0), L=(0,5), T=(3,7), R=(6,5)

2STEP 2

Cut along horizontal LR into a top triangle △TLR and a bottom triangle △BLR; both share base LR = 6 inches.

base = LR = 6 in

3STEP 3

Height to base y=5: top apex T is 7-5 = 2 above, bottom apex B is 5-0 = 5 below.

h_top = 7-5 = 2, h_bot = 5-0 = 5

4STEP 4

Two triangle areas $\frac{1}{2}$ (6)(2) = 6 and $\frac{1}{2}$ (6)(5) = 15; add them: 6 + 15 = 21.

\frac{1}{2}

(6)(2) +

\frac{1}{2}

(6)(5) = 6 + 15 = 21 → (A)

Answer

Two checks. (1) Bounding box: the kite fits inside the 6 × 7 = 42 rectangle from (0,0) to (6,7), and a kite typically fills about half of its bounding rectangle — that predicts

\frac{42}{2}

= 21, matching the answer. (2) Kite-diagonal formula: the diagonals are BT (length 7) and LR (length 6), and

\frac{1}{2}

d₁d₂ =

\frac{1}{2}

· 7 · 6 = 21 — same answer from a totally different route.

💡Key takeaway

Read the corners off the grid, slice the kite into two triangles along the horizontal y=5 line, and add: $\frac{1}{2}$ (6)(2) + $\frac{1}{2}$ (6)(5) = 21 — answer (A). A Grade 6 "decompose to find area" move handles an AMC 8 geometry problem.

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