AMC 8 · 2006 · #19

Grade 8 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

isosceles-trianglesimilar-triangleslogical-deduction identify-subproblems ↑ Prerequisites: isosceles-triangle

📏 Short solution 💡 3 insights 📊 Diagram

Problem

Isosceles △ ABC has AB = BC. Point D is the midpoint of both BC and AE. CE = 11, and △ ABD ≅ △ ECD. Find BD.

Pick an answer.

(A)

(B)

4.5

(C)

(D)

5.5

(E)

AMC 8 2006 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 natural opening: redraw the figure and mark every piece of information — the two equal sides AB = BC, the two midpoint splits at D, and the known length CE = 11. Tool #12 (Use Symmetry) reads the congruence △ ABD ≅ △ ECD off the vertex order: the matching pairs are A ⇔ E, B ⇔ C, D ⇔ D, so AB matches EC. That single matching turns the given CE = 11 into AB = 11, and the isosceles condition plus the midpoint cuts BC exactly in half. No algebra is needed — three sentences of side-chasing finish it.

1STEP 1

Redraw △ ABC with AB = BC, place E so D is the midpoint of both BC and AE, and label the one known length CE = 11.

AB = BC, BD = DC, AD = DE, CE = 11

2STEP 2

The vertex order in △ ABD ≅ △ ECD pairs A–E, B–C, D–D, so side AB matches side EC — giving AB = 11.

△ ABD ≅ △ ECD → AB = EC = 11

3STEP 3

The triangle is isosceles with AB = BC, and AB = 11, so BC = 11 too.

AB = 11 and AB = BC → BC = 11

4STEP 4

D is the midpoint of BC, so BD is half of BC: BD = $\frac{11}{2}$ = 5.5.

BD =

\frac{1}{2}

BC =

\frac{11}{2}

= 5.5 → (D)

Answer

5.5

Sanity-check the corresponding parts of △ ABD ≅ △ ECD. Side BD in the first triangle should match CD in the second, and indeed both equal 5.5 because D is the midpoint of BC. Side AD should match ED, which is exactly what the second midpoint condition gives. And AB = EC = 11 is what we used. All three pairs check out, so the congruence is consistent with the given midpoint conditions, and BD = 5.5 — answer (D). The other choices (4, 4.5, 5, 6) all force BC ≠ 11, contradicting the chain AB = EC = 11.

💡Key takeaway

Congruent triangles match corner-to-corner in the order they are written. Once AB = EC = 11, the isosceles side BC is also 11, and the midpoint cuts it in half.

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