AMC 8 · 2022 · #18

Grade 6 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Lattice or Non-Aligned Polygon Area

coordinate-geometryarea-rectanglesspatial-visualization coordinate-geometryidentify-subproblems ↑ Prerequisites: coordinate-geometryarea-rectangles

📏 Medium solution 💡 3 insights

Problem

A rectangle has its four side-midpoints located at (-3,0), (2,0), (5,4), and (0,4). Using only these four midpoints (not the rectangle's corners), find the area of the rectangle.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The problem is purely spatial — four points on a grid — so Tool #1 (Draw a Diagram) is the natural first move: plot the four midpoints on graph paper and the shape jumps out. The picture turns the puzzle into two clean subproblems (Tool #7): first find the area of the easy inner parallelogram formed by the midpoints, then relate it to the rectangle's area. To discover that ratio without memorizing a theorem, Tool #9 (Solve an Easier Related Problem) is perfect — try a simple axis-aligned rectangle whose midpoints are easy to write down, see that its midpoint-rhombus has exactly half the rectangle's area, and reuse that ratio here.

1STEP 1

Plot the four midpoints and join them in order A→B→C→D; sides AB and DC lie flat on y=0 and y=4, framing a tilted inner parallelogram.

A=(-3,0), B=(2,0), C=(5,4), D=(0,4)

2STEP 2

Base AB has length 5 on y=0 and the opposite side DC sits at y=4, so the inner parallelogram's area is 5 × 4 = 20.

Area(ABCD) = 5 × 4 = 20 square units

3STEP 3

Test an easy 6 × 4 rectangle: its midpoints form a rhombus of area 12, exactly half its area 24 — so the midpoint shape is always half.

midpoint area =

\frac{1}{2}

× 6 × 4 = 12 =

\frac{1}{2}

×(6× 4)

4STEP 4

The inner parallelogram is 20 and that is half the rectangle, so the rectangle's area is 2 × 20 = 40 → (C).

Area of rectangle = 2 × 20 = 40 → (C)

Answer

Look at the picture: the four midpoints stretch from x=-3 to x=5 (width 8) and from y=0 to y=4 (height 4), so they fit inside a bounding box of area 8 × 4 = 32. The actual rectangle is tilted, but its area should be a bit larger than the inner parallelogram (20) and on the same order as the bounding box. 40 sits comfortably between those — and it matches choice (C). Choices (A) 20 would mean the rectangle equals the inner parallelogram (impossible), and (E) 80 would mean the rectangle is four times bigger, which would burst past the bounding box.

💡Key takeaway

This AMC 8 problem only needs Grade 6 coordinate-geometry area skills you already know — plot the points, find the inner shape's area, double it!

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