Source: Revised GRE PDF 2nd Ed. Section 5; #4 (p. 76)

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# Three circles with their centers on line segment

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. Quantity A: The circumference of the largest circle Quantity B: The sum of the circumferences of the two smaller circles Quantity A is greater., Quantity B is greater., The two quantities are equal., The relationship cannot be determined from the information given.

### 6 Explanations

1

Sunmeet Singh Sethi

I did it in the following way:
D denotes the diameter of the larger circle, D1 denotes the diameter of the smaller circle and D2 denotes the diameter of the other smaller circle.
So, the circumference of the larger circle is (pi)D and circumference of the smaller circles is (pi)D1 and (pi)D2.
Dividing both sides with (pi) leaves us with D and D1+D2.
And since D1 and D2 are on the same line and add up to form the diameter of larger circle, hence the sum is equal to D.

Is this correct?

Oct 5, 2019 • Comment

Sam Kinsman

Yes, that's right! :D

3

P P

Why don’t we add two Pi together so it’s 6 pi squared

Oct 29, 2018 • Comment

Hey there, I take it you are referring to this calculation:

4π + 2π = 6π

We can first prove this mathematically. Let's say that π = 3, just for simplicity's sake. Thus:

4(3) + 2(3) = 6(3)
12 + 6 = 18

If we square the 3, then the two sides won't be equal anymore. The reason for this is that 4π and 2π are "like terms," in that they differ only in their coefficients (the 4 and the 2). When we add like terms, we add only the coefficients. So 2x + 3x = 5x.

This is because both terms are multiplied by the same thing, and then they are added. If they were being multiplied, then we would end up with π squared. But because we're dealing with addition, we don't want to multiply the π's together.

The expression 4π + 2π means that we have 4 "groups" of π, and 2 "groups" of π. When we add them, then, we'll have 6 "groups" of π.

1

Darnell Billups

i used equations to prove this, however, plugging numbers seems to go faster.
Circumference Circle PQ = (line PQ)(pi)
combined circumference of two smaller circles = (line PR)(pi) + (line RQ)(pi)
Factor pi: (pi)(line PR+lineRQ)
line PQ is line PR+ line RQ, which makes them equal because both equations have pi in them.

Oct 24, 2016 • Comment

Sam Kinsman

Yes, that's right! Good job :D

1

The question never says that PQ is the diameter however. It could be a line segment less than the diameter, which is why I chose D incorrectly. It just calls PQ a "line segment". Please advise! Does the mention of "tangent" mean that PQ is necessarily the diameter?

Jan 8, 2015 • Comment

Jonathan , Magoosh Tutor

Hi there! I think it's good that you didn't automatically assume the PQ is the diameter just by looking at the diagram.

However, the question says "three circles with their centers on line segment PQ," which tells us that PQ goes through the center of all three circles. Therefore, PQ must be the diameter of the largest circle; if it weren't the diameter, then it couldn't go through the center! I hope this clarifies :)

2

Dannialles Dominguez

What does it mean that the points P, Q, R are tangent points? The thing that made me skeptical in choosing numbers for this problem is that even though the circles seem to be next to each other, they could not be. In other words , the space where it seems they meet at R could have a gap between them and therefore the two quantities would not be equal. How do I know for sure that this is not the case?

Jan 20, 2014 • Comment

Hi there! I think you might have answered your own question. Saying that the points P, Q, and R tangent, means that the circles touch at these points. So there is not gap or space between the circles. This is definitely a good thing to keep in mind because sometimes the picture can be misleading, but with the information about tangent points, then we can safely assume that the picture is accurate. :)

3

Chris Lele

Sep 27, 2012 • Comment

Robert Paul

I would have gotten this wrong because I would think every time you add the circumference of multiple circles you'd do (4pi + 2pi) or (7pi + 3pi) or in other words (4 + 3.14 + 2 + 3.14) or (7 + 3.14 + 3 + 3.14). I'd add additional pi's. Good to know!