Problem:
Rhombus ABCD is similar to rhombus BFDE. The area of rhombus ABCD is 24, and ∠BAD=60∘. What is the area of rhombus BFDE?
Answer Choices:
A. 6
B. 43
C. 8
D. 9
E. 63
Solution:
Since ∠BAD=60∘, isosceles △BAD is also equilateral. As a consequence, △AEB,△AED,△BED,△BFD,△BFC, and △CFD are congruent. These six triangles have equal areas and their union forms rhombus ABCD, so each has area 24/6=4. Rhombus BFDE is the union of △BED and △BFD, so its area is (C)8.
OR
Let the diagonals of rhombus ABCD intersect at O. Since the diagonals of a rhombus intersect at right angles, △ABO is a 30−60−90∘ triangle. Therefore AO=3⋅BO. Because AO and BO are half the length of the longer diagonals of rhombi ABCD and BFDE, respectively, it follows that
