AMC 8 · 2001 · #11

Grade 6 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Axis-Aligned Figure Area from Coordinates

coordinate-geometryarea-rectanglesarea-triangles coordinate-geometryidentify-subproblems ↑ Prerequisites: coordinate-geometryarea-triangles

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Four points are plotted on the coordinate plane: A(3,2), B(3,-2), C(-3,-2), and D(-3,0). Connect them in order to form quadrilateral ABCD and find its area.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

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.

1STEP 1

Plot the four points and join them A → B → C → D → A: only side DA is slanted, so ABCD comes out as a trapezoid.

A(3,2), B(3,-2), C(-3,-2), D(-3,0)

2STEP 2

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.

Rectangle: (-3,-2),(3,-2),(3,0),(-3,0) Triangle: (-3,0),(3,0),(3,2)

3STEP 3

Rectangle area: width 6 (x=-3 to x=3) times height 2 (y=-2 to y=0) gives 12.

A_rect = 6 × 2 = 12

4STEP 4

Right triangle area: half of base 6 ((-3,0) to (3,0)) times height 2 (up to (3,2)) gives 6.

A_tri =

\frac{1}{2}

× 6 × 2 = 6

5STEP 5

The two pieces meet only along the cut and never overlap, so add them for 18.

A_ABCD = A_rect + A_tri = 12 + 6 = 18 → (C)

Answer

Bounding-box check: the smallest rectangle that contains all four points spans x ∈ [-3,3] and y ∈ [-2,2], an area of 6 × 4 = 24. Our answer 18 sits below 24, as it must, and the missing piece is a right triangle with legs 6 and 2 (area 6). Indeed 24 - 6 = 18. The two views agree, so the answer is consistent.

💡Key takeaway

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.

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