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Another method to answer this geometry question ?

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Another method to answer this geometry question ?

by aalradadi » Mon Dec 16, 2013 2:33 pm
"145. The figure shown above consists of three identical circles that are tangent to each other. If the area of the shaded region is , what is the radius of each circle?

(A) 4
(B) 8
(C) 16
(D) 24
(E) 32"

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by GMATGuruNY » Mon Dec 16, 2013 3:14 pm
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The figure shown consists of three identical circles that are tangent to each other. If the area of the shaded region (center space where the three circles do not touch) is (64)(square root 3) - 32 pi, what is the radius of each circle?

a. 4
b. 8
c. 16
d. 24
e. 32

OA: B
Shaded region = triangle - 3 circle sectors = 64√3 - 32π.

Looking at the equation above, we can see that the red portions must be equal:
3 circle sectors = 32Ï€.

Since the triangle is equilateral, each of its angles is 60 degrees.
Since 60/360 = 1/6, each sector is 1/6 the area of a circle.
Thus, the 3 sectors = 3*(1/6) = 1/2 the area of a circle.
Since the 3 sectors = 32Ï€, each circle area = 64Ï€.
Thus:
πr² = 64π
r=8.

The correct answer is B.
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by Jeff@TargetTestPrep » Fri Jan 05, 2018 9:15 am
aalradadi wrote:"145. The figure shown above consists of three identical circles that are tangent to each other. If the area of the shaded region is , what is the radius of each circle?

(A) 4
(B) 8
(C) 16
(D) 24
(E) 32
We can let each radius = r, and so the side of each triangle = 2r.

Notice that the area of the equilateral triangle consists of the central shaded region and three identical circular sectors, each of which is a 60-degree sector from its circle. Using this information, we can create the following equation:

(Area of equilateral triangle) - (3 x area of 1/6 of each circle) = area of shaded region

(2r)^2√3/4 - 3(1/6 x π r^2) = 64√3 − 32π

[(4r^2)√3]/4 - (πr^2)/2 = 64√3 − 32π

(r^2)√3 - (πr^2)/2 = 64√3 − 32π

Multiplying both sides by 2, we have:

2(r^2)√3 - πr^2 = 128√3 − 64π

r^2(2√3 - π) = 128√3 − 64π

Dividing both sides by (2√3 - π), we have:

r^2 = 64

r = 8

Answer: B

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