BTGmoderatorDC wrote:

If each side of ΔACD above has length 3 and if AB has length 1, what is the area of region BCDE?

(A) \(\frac{9}{4}\)

(B) \(\frac{7}{4} \sqrt{3}\)

(C) \(\frac{9}{4} \sqrt{3}\)

(D) \(\frac{7}{2} \sqrt{3}\)

(E) \(6 + \sqrt{3}\)

OA B

Source: Official Guide

Given each side of ΔACD above has length 3, ∆ACD is an equilateral triangle, thus, /_A = 60º. It is given that /_ABE = 90º, thus, /_ABE = 30º. Thus, ∆ABE is a 90-60-30 triangle. In a 90-60-30 triangle, the ratio of sides opposite to respective angles is 2 : √3 : 1. Given that AE, the side opposite to /_30º is 1, thus, the side opposite to /_60º would be √3. Thus, the area of ∆ABE = 1/2*AB*AE = 1/2*1*√3 = √3/2.

Note that the area of an equilateral triangle = √3/4*side^2 = √3/4*3^2 = 9√3/4

Thus, the area of region BCDE = 9√3/4 - √3/2 = 9√3/4 - 2√3/4 = 7√3/4

The correct answer: B

Hope this helps!

-Jay
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