AMC 10 · 2012 · #19

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Line Intersection Vertices then Area
coordinate-geometrysimilar-trianglesarea-triangles convert-to-algebra ↑ Prerequisites: coordinate-geometry
📏 Medium solution 💡 3 insights
Problem
A rectangle ABCD has AB=6 and AD=30. Point G is the midpoint of side AD. Side AB is extended past B by 2 units to a point E, and F is where segment ED crosses side BC. Find the area of the four-sided region BFDG.

Pick an answer.

(A)
$\frac{133}{2}$
(B)
67
(C)
$\frac{135}{2}$
(D)
68
(E)
$\frac{137}{2}$

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The figure mixes a rectangle, an extended side, and a crossing line, so the cleanest move is to pin it onto a coordinate grid where every point gets a clear address. Then the work splits into two subproblems: first locate F on side BC, then measure the area of BFDG. Once the points are placed, BFDG turns out to be a trapezoid, so its area is a single formula.

1STEP 1

Put the rectangle on a grid

Put A at the origin: A=(0,0), B=(6,0), C=(6,30), D=(0,30), G=(0,15) is the midpoint of AD, and E=(8,0) sits 2 past B.

A=(0,0), B=(6,0), C=(6,30), D=(0,30), G=(0,15), E=(8,0)
2STEP 2

Find F with similar triangles

Triangles EBF and EAD share angle E and have parallel bases, so BF is EBEA=28\frac{EB}{EA}=\frac{2}{8} of AD=30: BF=152\frac{15}{2}.

BF/AD=EB/EA=2/8=1/4 → BF=30/4=15/2
3STEP 3

See BFDG as a trapezoid

BF and GD sit on the vertical lines x=6 and x=0, so BFDG is a trapezoid with parallel sides 152\frac{15}{2} and GD=15, height 6.

BF=15/2, GD=15, height=6
4STEP 4

Apply the trapezoid area formula

Average the parallel sides and multiply by the height: 12(152+15)⋅6=\frac{1}{2}(\frac{15}{2}+15)\cdot 6= 1352\frac{135}{2}, choice (C).

Area=1/2(BF+GD) · h=1/2(15/2+15) · 6=135/2
Answer
135/2
The whole rectangle has area 6 × 30=180. Our trapezoid area 135/2=67.5 is a bit over a third of that, which looks right for a region spanning the full height on one side and part of it on the other. A quick Shoelace check on B(6,0),F(6,7.5),D(0,30),G(0,15) also gives 1/2|135|=135/2, confirming the answer.
💡Key takeaway

Drop the shape onto a grid, use similar triangles to find the missing corner, and the region becomes a trapezoid you can measure with one formula.

  • Put the rectangle on a grid
  • Find F with similar triangles
  • See BFDG as a trapezoid
  • Apply the trapezoid area formula

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