AMC 10 · 2006 · #13

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Parallel-Cut Area Ratio
similar-figuresequilateral-triangleratio-proportion symmetry-argument ↑ Prerequisites: similar-figures
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
A rhombus of area 24 has a sixty degree angle. A similar rhombus shares its shorter diagonal. Find the area of the second rhombus.

Pick an answer.

(A)
6
(B)
$4\sqrt {3}$
(C)
8
(D)
9
(E)
$6\sqrt {3}$
How to solve
Strategy Draw a Diagram

The whole problem lives in one figure, so Tool #1 (Draw a Diagram) is primary: reading the rhombus tells you that BD is a diagonal of both rhombi and that the 60° angle turns triangle ABD into an equilateral triangle. Tool #7 (Identify Subproblems) splits the work into two smaller jobs — first pin down both diagonals of ABCD from its area, then compare the two rhombi. Tool #4 (Introduce a Variable) names the common side length s so every diagonal is written in terms of one letter, which is exactly what makes the shared diagonal BD link the big rhombus to the small one.

1STEP 1

The 60 degree angle makes an equilateral triangle

The sixty degree angle makes an equilateral triangle inside.

AB=AD=s, ∠ BAD=60° → △ ABD equilateral → BD=s
2STEP 2

Find the long diagonal and the area

That gives both diagonals and ties the area to the side squared.

AC=s√3, Area_ABCD=1/2 BD · AC=√3/2s²=24
3STEP 3

Locate BD inside the smaller rhombus

In the smaller rhombus the shared diagonal is the long one.

t√3=BD=s → t=s/√3 → t²=s²/3
4STEP 4

Compute the area of BFDE

That makes its area one third, giving 8, choice (C).

Area_BFDE=√3/2t²=1/3·√3/2s²=1/3 · 24=8 → (C)
Answer
8
The answer 8 is exactly one third of the given area 24, which fits the picture: the inner rhombus BFDE is clearly smaller than ABCD, and a third of the area is a believable amount of shrink. It also passes the similarity test — because area scales as the square of the linear ratio and the side ratio is t/s=1/√3, the area ratio is (1/√3)²=1/3, matching 24→ 8. The distractors 9, 6, 6√3, and 4√3 do not come from this clean 1/3 scaling.
💡Key takeaway

When two shapes are similar and share a segment, figure out whether that segment is the long or the short diagonal in each — here BD is short for the big rhombus but long for the small one, which makes the small area exactly one third, so 8.

  • The 60 degree angle makes an equilateral triangle
  • Find the long diagonal and the area
  • Locate BD inside the smaller rhombus
  • Compute the area of BFDE

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