Competition · AMC preparation · step 4 of 4
AMC 8 · 2022 · #18
Grade 6 geometry-2dPick an answer.
AMC 8 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Plot the four midpoints
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.
Plotting four ordered pairs on a coordinate grid is exactly the Grade 5 coordinate-graphing skill.
5.G.A.2Draw A DiagramFind the parallelogram area
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.
Subproblem #1: use coordinates to read off the parallelogram's base and height directly from the plot — a Grade 6 coordinate-polygon move.
6.G.A.3Identify SubproblemsTest an easier rectangle
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.
Trying a simple axis-aligned rectangle shows the 1/2 ratio with no theorem needed — a Grade 6 composing/decomposing argument.
The quadrilateral formed by joining the four side-midpoints of the rectangle has exactly half the rectangle's area.
▸ Why?
Drawing the two lines that join the midpoints of opposite sides cuts the rectangle into four equal smaller rectangles, and inside each one the midpoint quadrilateral covers exactly half.
▸ Why?
Those two midlines meet at the center and split the rectangle into four smaller rectangles of the same size, so the four pieces fitted back together are the whole rectangle.
▸ Why?
In each small rectangle one edge of the midpoint quadrilateral runs corner to corner as its diagonal, cutting that small rectangle into two equal triangles, and the triangle on the inner side belongs to the midpoint shape.
▸ Why?
A rectangle's diagonal splits it into two triangles that turn onto each other exactly around the center, so the two triangles have equal area — half of the small rectangle each.
Double to get the rectangle
The inner parallelogram is 20 and that is half the rectangle, so the rectangle's area is 2 × 20 = 40 → (C).
Subproblem #2: combine the inner area with the rectangle-to-midpoint ratio to get the answer — Grade 6 area composition.
6.G.A.1Identify SubproblemsThis 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!
- Plot the four midpoints
- Find the parallelogram area
- Test an easier rectangle
- Double to get the rectangle
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