Competition · AMC preparation · step 4 of 4
AMC 8 · 2000 · #25
Grade 6 geometry-2d
Pick an answer.
AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Draw the rectangle
Draw rectangle ABCD, mark M on BC and N on CD, and pick a friendly area-72 rectangle: AB = 12, BC = 6.
A clean rectangle on a grid makes every length a whole number, so each corner triangle's area is just a quick base-times-height-divided-by-two.
5.G.A.1Draw A DiagramRead the four lengths
M is the midpoint of BC, so BM = MC = 3; N is the midpoint of CD, so CN = ND = 6.
"Midpoint" simply means "half the side" — once labeled on the picture, the legs of each corner triangle are obvious.
5.G.A.1Draw A DiagramFind the corner triangles
Each corner right triangle has both legs on the rectangle, so its area is easy: [ABM] = 18, [MCN] = 9, [AND] = 18.
Right triangles with legs on the rectangle's sides are the easiest area to compute — that's why subtracting them is the right move.
6.G.A.1Solve An Easier Related ProblemSubtract from the rectangle
The three corner triangles cover everything outside △ AMN, so subtract them from the rectangle: [AMN] = 72 - 45 = 27 → (B).
When the target shape is hard to measure but its surroundings are easy, subtract the surroundings from the whole.
The area of △ AMN is what is left after the three corner right triangles △ ABM, △ MCN, and △ AND are removed from rectangle ABCD.
▸ Why?
Triangle AMN and the three corner triangles fit together to fill the whole rectangle, with no gap between them and no piece overlapping another.
▸ Why?
The three drawn segments AM, MN, and NA cut the rectangle's inside into exactly four regions — the middle triangle and one triangle in each of the corners B, C, D — and every point of the rectangle sits in exactly one of them, so the four areas add back to the rectangle's area.
▸ Why?
Because those four areas add up to the rectangle, taking the three corner areas away from the rectangle's area leaves exactly the fourth area, △ AMN — subtracting undoes the adding.
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.
- Draw the rectangle
- Read the four lengths
- Find the corner triangles
- Subtract from the rectangle
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