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555-605 (Medium)|   Geometry|      
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Bunuel
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Three circles with their centers on line segment PQ are tangent at poi 19 May 2019, 23:15
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FRESH GMAT CLUB TESTS QUESTION




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 \(14\pi\)
(2) The ratio of the radii of two smaller circles is 4:3


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M36-122

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Bunuel
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Three circles with their centers on line segment PQ are tangent at poi 16 Nov 2021, 04:08
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Bunuel

FRESH GMAT CLUB TESTS QUESTION




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 \(14\pi\)
(2) The ratio of the radii of two smaller circles is 4:3


Show Spoiler
M36-122

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72588.png
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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
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Three circles with their centers on line segment PQ are tangent at poi 19 May 2019, 23:22
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π (d1 + d2) = 14 π

d1 + d2 = D (diameter of the largest circle) = 14

Circumference hence is 14π.

A. alone is sufficient to answer.

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Three circles with their centers on line segment PQ are tangent at poi 19 May 2019, 23:34
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Bunuel

FRESH GMAT CLUB TESTS QUESTION




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 \(14\pi\)
(2) The ratio of the radii of two smaller circles is 4:3


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Ans A

Let's assume that the radius of the smaller circus (on the left) is r1 and radius of the second smaller circus (on the right) is r2
From the question stem we know that: r1 + r2 = radius of the largest circus (R).

In order to calculate the circumference of the largest circle we need to know its radius (R).

Stm 1: 2πr1 + 2πr2 = 14π
2π(r1+r2)=14π
r1+r2=7= R (radius of the largest circle)

SUFFICENT to calculate circumference of the largest circle

Stm2: r1/r2= 4/3
r1= (4/3)r2
r1+r2 = (4/3)r2 + r2 = (7/3)r2

We have one equation with two unknown variables.

NOT SUFFICIENT to calculate circumference of the largest circle

Hope it helps
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Three circles with their centers on line segment PQ are tangent at poi 19 Oct 2019, 06:19
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Bunuel

FRESH GMAT CLUB TESTS QUESTION




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 \(14\pi\)
(2) The ratio of the radii of two smaller circles is 4:3


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Attachment:
72588.png
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Circle PQ: a, Circle PR: b, Circle RQ: c
Circumference = diameter•π = 2•radius•π

(1) The sum of the circumferences of the two smaller circles is \(14\pi\): sufic.

\(b_{circ}+{circ}=14π… 2b_dπ+2c_dπ=14π…b_d+c_d=14\)
\(a_{diameter}=b_d+c_d…a_d=14\)
\(a_{circ}=a_{diameter}(π)…a_{circ}=14π\)

(2) The ratio of the radii of two smaller circles is 4:3 insufic.

\(b_r/c_r=4/3…b_d=8x…c_d=6x…a_d=14x\)
\(a_{circ}=14x(π)…a_{circ}=[14π,28π,etc…]\)

Answer (A)
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Three circles with their centers on line segment PQ are tangent at poi 18 Jan 2023, 12:51
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