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Three circles with their centers on line segment PQ are tangent at points P, R, and Q, where point R lies on line segment PQ. What is the circumference of the largest circle?
(1) The sum of the circumferences of the two smaller circles is \(14π\)
(2) The ratio of the radii of two smaller circles is \(4:3\)
16 Nov 2021, 04:07
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Official Solution:
Three circles with their centers on line segment PQ are tangent at points P, R, and Q, where point R lies on line segment PQ. What is the circumference of the largest circle?
Notice that since the diameter of the largest circle equals to the sum of the diameters of the two smaller circles, then \(d_1+d_2=D\). Divide by 2: \(r_1+r_2=R\) (the radius of the largest circle equals to the sum of the radii of the two smaller circles).
(1) The sum of the circumferences of the two smaller circles is \(14π\)
The above means that that \(2\pi r_1+2\pi r_2=14π\).
Reduce by \(2\pi\) to get \(r_1+r_2=7\). We know the radius of the largest circle, so we can get the circumference. Sufficient.
(2) The ratio of the radii of two smaller circles is \(4:3\)
We know only that ratio and nothing about the actual length of anything. Not sufficient.
Answer: A
Three circles with their centers on line segment PQ are tangent at points P, R, and Q, where point R lies on line segment PQ. What is the circumference of the largest circle?
Notice that since the diameter of the largest circle equals to the sum of the diameters of the two smaller circles, then \(d_1+d_2=D\). Divide by 2: \(r_1+r_2=R\) (the radius of the largest circle equals to the sum of the radii of the two smaller circles).
(1) The sum of the circumferences of the two smaller circles is \(14π\)
The above means that that \(2\pi r_1+2\pi r_2=14π\).
Reduce by \(2\pi\) to get \(r_1+r_2=7\). We know the radius of the largest circle, so we can get the circumference. Sufficient.
(2) The ratio of the radii of two smaller circles is \(4:3\)
We know only that ratio and nothing about the actual length of anything. Not sufficient.
Answer: A
