"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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Another method to answer this geometry question ?
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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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We can let each radius = r, and so the side of each triangle = 2r.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
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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