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 $BFDE?$

$6$

$4\sqrt{3}$

$8$

$9$

$6\sqrt{3}$

Answer: C

Concepts:rhombus equilateral triangle area decomposition

Difficulty rating: 1660

Solution:

Let the diagonals of $ABCD$ meet at $O.$ They bisect each other at right angles, and since $\angle BAD=60^\circ,$ triangle $ABO$ is a $30^\circ$ – $60^\circ$ – $90^\circ$ triangle. Hence the half-diagonals satisfy $AO=\sqrt3\,BO.$

The segment $BD$ is the short diagonal of $ABCD$ and the long diagonal of the similar rhombus $BFDE.$ Thus the smaller-to-larger length ratio is $BD/AC=BO/AO=1/\sqrt3.$ Areas scale by the square of this ratio, so the area of $BFDE$ is $24\cdot\dfrac13=8.$

Thus, the correct answer is C.

2006 AMC 12B Problem 13

Answer: C

Difficulty rating: 1660

Solution:

Problem 13 in Other Years