AMC 8 · 2000 · #25

Grade 6 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation
area-rectanglesarea-trianglescoordinate-geometryfraction-multiplication area-differenceidentify-subproblemscoordinate-geometry ↑ Prerequisites: area-rectanglesarea-triangles
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
Rectangle ABCD has area 72. Let M be the midpoint of BC and N be the midpoint of CD. Connect A, M, and N to form a triangle. Find the area of △ AMN.

Pick an answer.

(A)
21
(B)
27
(C)
30
(D)
36
(E)
40

AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Change Focus / Count the Complement

Computing the area of △ AMN directly is awkward because none of its sides are horizontal or vertical. Tool #16 (Count the Complement) flips the problem: A, M, N cut three corner right triangles out of rectangle ABCD, and the inside triangle is whatever is left. Each corner triangle has legs along the rectangle's sides, so its area is easy. Tool #1 (Draw a Diagram) makes the three corner pieces visible. Tool #9 (Easier Related Problem) lets us pick a convenient rectangle with area 72 — say 12 × 6 — to keep arithmetic with whole numbers; the answer for a midpoint construction depends only on the area, not the specific shape.

1STEP 1

Draw rectangle ABCD, mark M on BC and N on CD, and pick a friendly area-72 rectangle: AB = 12, BC = 6.

AB = 12, BC = 6, area(ABCD) = 12 × 6 = 72
2STEP 2

M is the midpoint of BC, so BM = MC = 3; N is the midpoint of CD, so CN = ND = 6.

BM = MC = 3, CN = ND = 6
3STEP 3

Each corner right triangle has both legs on the rectangle, so its area is easy: [ABM] = 18, [MCN] = 9, [AND] = 18.

[ABM] = 18, [MCN] = 9, [AND] = 18
4STEP 4

The three corner triangles cover everything outside △ AMN, so subtract them from the rectangle: [AMN] = 72 - 45 = 27 → (B).

[AMN] = 72 - (18 + 9 + 18) = 72 - 45 = 27 → (B)
Answer
27
Check the answer is independent of the rectangle's shape. Repeat with AB = 9 and BC = 8 (still area 72): BM = 4, CN = 4.5, AD = 8. Then [ABM] = 12\frac{1}{2}(9)(4) = 18, [MCN] = 12\frac{1}{2}(4)(4.5) = 9, [AND] = 12\frac{1}{2}(4.5)(8) = 18. Sum = 45, and [AMN] = 72 - 45 = 27. The same answer comes out — confirming the area only depends on the rectangle's area. A symbolic check makes this exact: if AB = l and BC = w, then [ABM] = lw/4, [MCN] = lw/8, [AND] = lw/4, summing to 5lw/8, and [AMN] = lw - 5lw/8 = 3lw/8 = 38\frac{3}{8}(72) = 27. Answer (B).
💡Key takeaway

The triangle AMN is tilted, so its area is hard to measure head-on. Flip the question: the three corner triangles ABM, MCN, AND sit on the rectangle's sides, so their areas are easy (18 + 9 + 18 = 45). Subtract from the rectangle's area: 72 - 45 = 27. Answer (B). The trick — "count what's around it instead of what you want" — works for any area 72, no matter the rectangle's shape.

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