AMC 10 · 2011 · #9

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
similar-trianglesarea-trianglesinteger-pythagorean-triples identify-subproblems ↑ Prerequisites: area-triangles
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
In the figure, triangle ABC has side lengths AC=3, CB=4, and AB=5. Point D lies on AB and point E lies on CB, and segment DE is perpendicular to AB. The area of △ EBD is exactly one third of the area of △ ABC. Find the length BD.

Pick an answer.

(A)
$\frac{4}{3}$
(B)
$\sqrt{5}$
(C)
$\frac{9}{4}$
(D)
$\frac{4\sqrt{3}}{3}$
(E)
$\frac{5}{2}$

AMC 10 2011 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 key: reading the figure shows that △ EBD and △ ABC share the corner at B and each contain a right angle (at D and at C), which is exactly the setup for similar triangles. Tool #7 (Identify Subproblems) breaks the work into three clean pieces — find the area of △ ABC, turn the area condition into a ratio of sides, then read off BD. Tool #4 (Introduce a Variable) names the shared scale factor k so the area-to-side relationship becomes a short equation.

1STEP 1

Recognize the 3-4-5 right triangle

The sides 3, 4, 5 satisfy 3²+4² = 25 = 5², so △ ABC is right-angled at C, opposite the longest side AB=5.

3² + 4² = 25 = 5² → ∠ C = 90°
2STEP 2

Area of the big triangle

The legs CA=3 and CB=4 give [△ ABC] = 1/2·3·4 = 6, so [△ EBD] = 1/3·6 = 2.

[△ ABC] = 1/2 · 3 · 4 = 6, [△ EBD] = 1/3 · 6 = 2
3STEP 3

Find the similar triangles

DE ⊥ AB makes ∠ BDE = 90° = ∠ BCA, and ∠ B is shared, so △ BDE ∼ △ BCA by AA; BD matches BC.

∠ BDE = ∠ BCA = 90°, ∠ B shared → △ BDE ∼ △ BCA
4STEP 4

Area ratio gives the scale factor

Areas scale as the square of the side ratio k = BD/BC, so k² = 2/6 = 1/3.

k² = ([△ EBD])/([△ ABC]) = 2/6 = 1/3
5STEP 5

Solve for BD

Square-rooting, k = 1/√3, so BD = 4·1/√3 = 4√3/3 — choice (D).

k = 1/√3, BD = 4·1/√3 = 4/√3 = 4√3/3 → (D)
Answer
4√(3)/3
BD = 4√3/3≈ 2.31, which is less than BC=4 and less than AB=5, so D falls comfortably between A and B — a sensible spot on the segment. Checking the area directly: DE = CA · k = 3·1/√3 = √3, so [△ EBD] = 1/2 · BD · DE = 1/2·4/√3·√3 = 2, exactly one third of 6. This confirms choice (D).
💡Key takeaway

Same-shape triangles have areas that grow by the square of the side ratio, so match the areas, take the square root, and scale the side you know.

  • Recognize the 3-4-5 right triangle
  • Area of the big triangle
  • Find the similar triangles
  • Area ratio gives the scale factor
  • Solve for BD

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