Competition · AMC preparation · step 4 of 4
AMC 8 · 2001 · #11
Grade 6 geometry-2d
Pick an answer.
AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Coordinates always invite Tool #1 (Draw a Diagram): plot the four points on grid paper and the quadrilateral's shape becomes obvious. Once we can see it, the slanted side DA is the only awkward feature, so we reach for Tool #7 (Identify Subproblems): cut the quadrilateral along the x-axis into a rectangle (below) and a right triangle (above). Both have horizontal and vertical sides, so each area is one easy Grade 6 formula. Add the two pieces. We deliberately skip Tool #13 (algebra/coordinate formulas like the Shoelace formula) because decomposition is faster and uses only elementary area formulas.
Plot the four points
Plot the four points and join them A → B → C → D → A: only side DA is slanted, so ABCD comes out as a trapezoid.
Plotting ordered pairs in the coordinate plane and reading off the resulting figure is the core Grade 5 coordinate-graphing skill.
5.G.A.2Draw A DiagramCut along the x-axis
Cut along the x-axis: it runs through D(-3,0) and (3,0) on side DA, splitting ABCD into a rectangle below and a right triangle above.
Decomposing a polygon into rectangles and right triangles is exactly the Grade 6 "find area by composing/decomposing" technique.
6.G.A.1Identify SubproblemsFind the rectangle's area
Rectangle area: width 6 (x=-3 to x=3) times height 2 (y=-2 to y=0) gives 12.
Width × height for a rectangle is the Grade 4 area formula.
4.MD.A.3Identify SubproblemsFind the triangle's area
Right triangle area: half of base 6 ((-3,0) to (3,0)) times height 2 (up to (3,2)) gives 6.
Half of base times height for a right triangle is the standard Grade 6 area rule.
6.G.A.1Identify SubproblemsAdd the two pieces
The two pieces meet only along the cut and never overlap, so add them for 18.
Area is additive over non-overlapping pieces — the heart of the decomposition standard.
The area of quadrilateral ABCD equals the area of the rectangle piece plus the area of the right-triangle piece.
▸ Why?
The x-axis runs through vertex D(-3,0) and through the point (3,0) on side AB, so one straight cut splits ABCD into a rectangle below the axis and a right triangle above it, with no part of the shape left out.
▸ Why?
The rectangle and the triangle cover every point of ABCD and meet only along the cut segment from (-3,0) to (3,0), so nothing is left uncovered and no region is counted twice.
Plot the points, then slice the shape into a rectangle and a right triangle — both areas are Grade 6 formulas, and their sum is the answer.
- Plot the four points
- Cut along the x-axis
- Find the rectangle's area
- Find the triangle's area
- Add the two pieces
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