Competition · AMC preparation · step 4 of 4

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

Label the square

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

Read the legs off the diagram

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

Find the triangle's area

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

[△ BEC] = 1/2 · BE · EC = 1/2 · 3 · 3 = 9/2 = 4.5 → (B)

Once the legs are known, the Grade 6 area formula closes the problem in one line.

Triangle BEC is right-angled at E with legs BE = 3 and EC = 3, so its area is half the product of those two legs.

▸ Why?

The corner of △ BEC at E is a right angle, because BE stands perpendicular to the base line DC.

▸ Why?

BE is parallel to AD, and DC crosses both of these parallel segments; a crossing line makes matching angles equal, so the right angle that AD forms with DC at D is copied where BE meets DC at E.

A transversal of parallel lines makes equal anglesA line crossing two parallel lines makes matching corresponding and alternate angles equal.

▸ Why?

Each leg measures 3: the upright leg BE matches side AD, and the flat leg EC is what remains of DC after taking away DE.

▸ Why?

BE is the segment AD slid straight sideways to pass through B and E, and sliding a segment without turning it keeps its length, so BE = AD = 3.

Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).

▸ Why?

E lies on DC, cutting it into DE and EC; since DE is 3 and the two pieces together fill DC = 6, the leftover EC is 3.

▸ Why?

DE is side AB slid straight down onto the base line, and sliding keeps length, so DE = AB = 3.

Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).

▸ Why?

DE and EC meet at E with no gap and no overlap, so their lengths add up to the whole length DC.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

The area of this right triangle is half the product of its two legs, because the triangle is exactly one half of the rectangle built on those legs.

▸ Why?

Build the rectangle with sides BE and EC; its diagonal BC cuts it into two triangles that lay exactly onto each other, so △ BEC is one of two equal halves of that rectangle.

Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).

6.G.A.1Identify Subproblems

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.

Label the square
Read the legs off the diagram
Find the triangle's area

A parent dashboard for the family lives at sensimlab.com.

Label the square

Read the legs off the diagram

Find the triangle's area

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