reto wrote:

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ABC is an equilateral triangle of area 3, and arc DE is centered at C. If E is the midpoint of AC, what is the area of the shaded region?

A. \(3 - \frac{\sqrt(3)*\pi}{2}\)

B. \(3 - \frac{\pi}{\sqrt(3)}\)

C. \(3 - \frac{\pi}{2}\)

D. \(3 - \frac{\pi}{2*\sqrt(3)}\)

E. \(3 - \frac{\pi}{6}\)

To approximate, you can use the property of similar triangles. If you join DE, you get that triangle DEC is similar to triangle ABC such that side of DEC is half of the side of ABC. So area of DEC will be 1/4 of the area of ABC (which is 3).

The shaded region is a little less than 3 - 3/4. All options are in the form of (3 - Something).

This 'something' would be slightly more than 3/4 but perhaps less than 1.

A. \(3 - \frac{\sqrt(3)*\pi}{2}\)

Something = 1.7*3.14/2 (much greater than 1)

Not possible

B. \(3 - \frac{\pi}{\sqrt(3)}\)

Something = 3.14/1.7 (much greater than 1)

Not possible

C. \(3 - \frac{\pi}{2}\)

Something = 3.14/2 (much greater than 1)

Not possible

D. \(3 - \frac{\pi}{2*\sqrt(3)}\)

Something = 3.14/2*1.7 = 3.14/3.4

Certainly possible

E. \(3 - \frac{\pi}{6}\)

Something = 3.14/6 = Almost 1/2

Not possible since it must be greater than 3/4

Answer (D)

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