Three circles with their centers on line segment PQ are tangent at poi

Question

FRESH GMAT CLUB TESTS QUESTION https://gmatclub.com/forum/download/file.php?id=47924 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 M36-122 72588.png

Bunuel · Accepted Answer

Official Solution:

https://gmatclub.com/tests-beta/resources/import/img/M36-122.png 
 
 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

Shorub · Answer

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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kebab1990 · Answer

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

exc4libur · Answer

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}=

Answer (A)

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Three circles with their centers on line segment PQ are tangent at poi

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