2006 AMC 10B Problem 15

Question

Rhombus

ABCD

is similar to rhombus

BFDE.

The area of rhombus

ABCD

is

24,

and

\angle BAD=60^\circ.

What is the area of rhombus

BFDE?

6

4\sqrt{3}

8

9

6\sqrt{3}

Since

AB=AD

and

\angle BAD=60^\circ,

triangle

ABD

is equilateral

E

and

F

split

ABCD

into six congruent triangles

Because

AB=AD

and

\angle BAD=60^\circ,

triangle

ABD

is equilateral, and so is triangle

CBD.

Points

E

and

F

split the rhombus into six congruent triangles, each of area

\tfrac{24}{6}=4.

Rhombus

BFDE

is the union of triangles

BED

and

BFD,

so its area is

2\cdot4=8.

Thus, the correct answer is C.

Problem 15 in other years