Problem:
Equilateral triangle DEF is inscribed in equilateral triangle ABC as shown with DE⊥BC. The ratio of the area of △DEF to the area of △ABC is
Answer Choices:
A. 61
B. 41
C. 31
D. 52
E. 21
Solution:
Since CDE is a right triangle with ∠C=60∘, we have CE=2DC. Also, ∠BFD=90∘=∠FEA. To see that ∠BFD=90∘, note that
∠BDF+∠FDE+90∘=∠BDF+60∘+90∘=180∘.
Thus ∠BDF=30∘ and since ∠DBF=60∘,∠BFD=90∘. That ∠FEA=90∘ follows similarly. Since △DEF is equilateral, the three small triangles are congruent and AE=DC. Let AC=3x. Then EC=2x and DE=3x. The desired ratio is
