AMC 8 · 2007 · #8

Grade 6 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation

area-trianglesarea-rectangles identify-subproblems ↑ Prerequisites: area-rectanglesarea-triangles

📏 Short solution 💡 1 insight 📊 Diagram

Problem

Trapezoid ABCD has AD ⊥ DC with AD = AB = 3 and DC = 6. Point E sits on DC so that BE ∥ AD. Find the area of △ BEC.

Pick an answer.

(A)

(B)

4.5

(C)

(D)

(E)

AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

Tool #1 (Draw a Diagram) is the lead because the answer pops out as soon as we annotate the figure: with AD ⊥ DC and BE ∥ AD, the piece ABED has two pairs of parallel sides and a right angle, so it is a rectangle — and since AD = AB = 3, it is a 3 × 3 square. Tool #7 (Identify Subproblems) then splits the trapezoid into the labeled square ABED plus the right triangle BEC, so the triangle's legs (BE and EC) read straight off the picture. No algebra needed.

1STEP 1

ABED has AB ∥ DE and AD ∥ BE, a right angle at D, and AD = AB = 3 — so it is a square of side 3.

ABED is a 3 × 3 square

2STEP 2

Since ABED is a square, BE = 3 and DE = 3, so EC = DC - DE = 6 - 3 = 3.

EC = DC - DE = 6 - 3 = 3

3STEP 3

△ BEC is right-angled at E with legs BE = 3 and EC = 3, so its area is ½ · 3 · 3 = 4.5 → (B).

[△ BEC] =

\frac{1}{2}

· BE · EC =

\frac{1}{2}

· 3 · 3 =

\frac{9}{2}

= 4.5 → (B)

Answer

4.5

Cross-check by computing the trapezoid's area two ways. Direct formula: [ABCD] =

\frac{1}{2}

(AB + DC) · AD =

\frac{1}{2}

(3 + 6) · 3 =

\frac{27}{2}

= 13.5. Decomposition: [square ABED] + [△ BEC] = 9 + 4.5 = 13.5. The two totals match, confirming the triangle's area is 4.5. Also, △ BEC is only a corner of the trapezoid, so an answer larger than the whole trapezoid (like (E) 18) would be impossible — 4.5 sits comfortably inside.

💡Key takeaway

When a trapezoid has one slanted side, drop a perpendicular from the top corner. The trapezoid splits into a rectangle (or square) plus a right triangle, and the triangle's legs read straight off the picture.

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