If the center of a circle lies on a smaller circle...
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If the center of a circle lies on a smaller circle, as shown above, what portion of the area of the larger circle is shaded?
A. 5/8
B. 2/3
C. 11/16
D. 3/4
E. 7/9
The OA is D.
I get the solution to this PS question as follow,
Radius of smaller circle = r2.
Radius of bigger circle = r1.
r1 = 2r2.
Portion of the area of the shaded region: pir2^2.
pi(2r2)^2 - pi r2^2 = 3/4 pi r1^2.
What portion of the area of the larger circle is shaded?
(3/4 pi r1^2)/(pi r1^2) = 3/4. Option C.
Experts, any suggestion? Thanks!
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Let the diameter of the larger circle = 4, implying that the diameter of the smaller circle = 2.
Area of the larger circle = πr² = π(2)² = 4π.
Area of the smaller circle πr² = π(1)² = π.
Area of the unshaded portion = larger circle - smaller circle = 4π - π = 3π.
Thus:
(unshaded portion)/(larger circle) = (3Ï€)/(4Ï€) = 3/4.
The correct answer is D.
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We can let the radius of the smaller circle = 1 and thus the radius of the larger circle = the diameter of the smaller circle = 2. The area of the smaller circle is π and the area of the larger circle is 4π. Therefore, the area of the shaded region is 4π - π = 3π and hence the area of the shaded region is 3π/4π = 3/4 of the area of the larger circle.
Answer: D
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