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Circle X and Y are connected at one side. Larger circle C surrounds both circles. Line segment W Z is the diameter of circle C. The line segment goes through all points of the other circles. The length of W X is 8 and the length of Y Z is 10.
Line segment WX is the radius of circle X, and line segment ZY is the radius of circle Y. Points W, X, C, Y, and Z are all on line segment WZ.
What is the area of circle C, which passes though points W and Z?
81Pi units squared
164Pi units squared
324Pi units squared
1296Pi units squared
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Let R be the radius of circle C. Put W at x = −R and Z at x = +R on the diameter WZ (so C is at 0). Then
- X is 8 units right of W: x_X = −R + 8,
- Y is 10 units left of Z: x_Y = R − 10.
The two smaller circles are tangent, so the distance between their centers is 8 + 10 = 18:
x_Y − x_X = (R − 10) − (−R + 8) = 2R − 18 = 18.
Thus 2R = 36 and R = 18.
Area of circle C = πR^2 = π(18^2) = 324π.
Answer: 324Pi units squared.
- X is 8 units right of W: x_X = −R + 8,
- Y is 10 units left of Z: x_Y = R − 10.
The two smaller circles are tangent, so the distance between their centers is 8 + 10 = 18:
x_Y − x_X = (R − 10) − (−R + 8) = 2R − 18 = 18.
Thus 2R = 36 and R = 18.
Area of circle C = πR^2 = π(18^2) = 324π.
Answer: 324Pi units squared.
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