AMC 10 · 2006 · #17

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Line Intersection Vertices then Area
coordinate-geometrysimilar-trianglesarea-triangles symmetry-argument ↑ Prerequisites: coordinate-geometry
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
Rectangle ADEH has a top side AD of length 3 and a height AH of length 2. Points B and C cut the top side AD into three equal parts, and points G and F cut the bottom side HE into three equal parts. Four slanted segments are drawn — A to F, B to E, C to H, and D to G — and they cross each other to carve out a small tilted quadrilateral WXYZ in the middle. Find the area of WXYZ.

Pick an answer.

(A)
$\frac{1}{2}$
(B)
$\frac{\sqrt{2}}{2}$
(C)
$\frac{\sqrt{3}}{2}$
(D)
$\sqrt{2}$
(E)
$\frac{2\sqrt{3}}{3}$

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The picture is a tangle of four crossing segments, so the safest move is to pin the whole thing onto a coordinate grid (Tool #1, Draw a Diagram, made exact with coordinates). Once every labeled point has an (x,y) address, each drawn segment becomes a simple line equation (Tool #4, Introduce a Variable — here the variables are x and y). Two of the lines slant one way and two slant the other, and the two slants turn out to be perpendicular, which already tells us WXYZ is a rectangle. Then finding each corner is just a small subproblem: solve where two lines cross (Tool #7, Identify Subproblems). Finally the area comes straight from the corner coordinates. Coordinates trade a confusing figure for four tidy pieces of algebra.

1STEP 1

Put the rectangle on a grid

Set corner H at the origin: E(3,0), A(0,2), D(3,2), and the trisection points are B(1,2), C(2,2), G(1,0), F(2,0).

H(0,0), E(3,0), A(0,2), D(3,2), B(1,2), C(2,2), G(1,0), F(2,0)
2STEP 2

Write the four slanted lines

AF: y=2-x and BE: y=3-x have slope -1; CH: y=x and DG: y=x-1 have slope +1, so the two families meet at right angles.

AF: y=2-x, BE: y=3-x, CH: y=x, DG: y=x-1
3STEP 3

Solve for the four corners

Setting each crossing pair of equations equal gives Z=(1,1), W=(3/2,3/2), Y=(3/2,1/2), X=(2,1).

Z(1,1), W (3/2,3/2), X(2,1), Y (3/2,1/2)
4STEP 4

Take the area from the diagonals

Diagonals WY and XZ are perpendicular with length 1 each, so the area is half their product: 1/2, choice (A).

WY=1, XZ=1, [WXYZ]=1/2 · WY · XZ=1/2 · 1 · 1=1/2 → (A)
Answer
1/2
A quick side-length check agrees: one side runs from Z(1,1) to W(3/2,3/2), length √((1/2)²+(1/2)²)=√2/2, and (√2/2)²=1/2 — the same area. The value is also sensible in scale: the whole rectangle has area 6, and the little middle square taking up just 1/2 of that (one-twelfth) looks right for the tiny region in the picture. The distractors are the traps of stopping too early — √2/2 is the side length, not the area, and the √3 choices would need a 30-60-90 angle that never appears here. Only (A) matches.
💡Key takeaway

When a figure is a mess of crossing lines, drop it onto a coordinate grid — then every line is an equation and every crossing point is just where two equations agree.

  • Put the rectangle on a grid
  • Write the four slanted lines
  • Solve for the four corners
  • Take the area from the diagonals

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