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Re: Three circles with their centers on line segment AB are ta [#permalink]
how can we conclude that F is center of big circle ??

Originally posted by Pranaygre on 12 Sep 2020, 23:51.
Last edited by Pranaygre on 13 Sep 2020, 08:49, edited 1 time in total.
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Re: Three circles with their centers on line segment AB are ta [#permalink]
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We don't conclude F is the centre of line AB as F can be anywhere on the line. What we make use of the fact is that -> AF + FB = AB (irrespective of where F is, this property stands. So, we get:

AF + FB = AB
2r1 + 2r2 = 2R (Based on info given)
r1 + r2 = R
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Re: Three circles with their centers on line segment AB are ta [#permalink]
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This is a real world problem so we can use actual numbers to plug and solve:

IT's given in the diagram that bigger circle's diameter passes through smaller circles so let the bigger diameter be 12
so the perimeter will be 2 pie r = 12 pie, since ACB accounts for half the perimeter it'll be 6 pie

Now 2 smaller circles will have combined D of 12 so it can be 6,6 or 8,4 or any other combo
hence half the perimeter would account to 6 pie
so Answer is C
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Three circles with their centers on line segment AB are ta [#permalink]
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1] There are tangents at points A, F and B

2] AB is a line segment

Based on these 2 given facts in the question, we can easily say that all three tangents that pass through A, F and B are parallel to each other.

These also draws a conclusion that line segment AB passes through centers of all three circles.

Hence,
AB is diameter of the bigger circle = \(D\)
AF is diameter of inner upper circle = \(d_1\)
FB is diameter of inner lower circle = \(d_2\)

But note that nowhere in the question is point F mentioned to be the center of bigger circle nor is it midpoint of line segment AB.

We know that,

AF + FB = AB
\(d_1 + d_2 = D\)

All the arcs are half the circumference of their respective circles (since they have the diameters as their chords)

Quantity A - arc ACB = \(\frac{πD}{2}\)

Quantity B - arc ADF + arc FEB = \(\frac{πd_1}{2} + \frac{πd_1}{2} = \frac{π}{2}(d_1 + d_2) = \frac{πD}{2}\)

Hence, Answer is C
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Re: Three circles with their centers on line segment AB are ta [#permalink]
Carcass wrote:
Attachment:
GRE The length of arc ACB.png


Three circles with their centers on line segment AB are tangent at points A, F, and B, where point F lies on line segment AB.


Quantity A
Quantity B
The length of arc ACB
The sum of the lengths of arcs ADF and FEB


A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.


Hi Carcass, could you help clarify that why is in this question and below question that both haven't given any radius value, however in arcs or circumference question it can be equal (C) but in Area question it can't be determined (D)? Could you help clarify? Thanks a bunch. :thumbsup: :please:

https://gre.myprepclub.com/forum/three- ... 22631.html
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Re: Three circles with their centers on line segment AB are ta [#permalink]
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I got what you mean

The stem is equal and in the second question we do not have the radius and the answer is D but here we do not have as well and the answer is C for a similar reasoning

But in the second we need the areas and the smaller circles could be bigger or smaller inside the big one.

he instead we do have

All the arcs are half the circumference of their respective circles (since they have the diameters as their chords)

Hence is not so important my point in blue in this first question. We only care about that the arcs are half-circle, regardless their wideness
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Three circles with their centers on line segment AB are ta [#permalink]
Carcass wrote:
I got what you mean

The stem is equal and in the second question we do not have the radius and the answer is D but here we do not have as well and the answer is C for a similar reasoning

But in the second we need the areas and the smaller circles could be bigger or smaller inside the big one.

he instead we do have

All the arcs are half the circumference of their respective circles (since they have the diameters as their chords)

Hence is not so important my point in blue in this first question. We only care about that the arcs are half-circle, regardless their wideness


Hi Carcass, thanks for your explanation.

My confusion here is as cirrcumference/arc/area are all depends on radius for calculation and as you rightly mentioned that radius could be different for different circles.
Therefore not sure why radius could be the same when it come to arcs and circumference? Shouldn't this vary too depends on radius value?
Or to clarify as you mentioned due to All the arcs are half the circumference of their respective circles (since they have the diameters as their chords) therefore regardless of radius value, arc/circumference value will always be half/same accordingly? Is my understanding correct?
Could you help clarify and thanks a bunch :please: :)
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Re: Three circles with their centers on line segment AB are ta [#permalink]
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