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# Three circles with radii of 1, 2, and 3, lie on the same plane. Do any

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Math Expert
Joined: 02 Sep 2009
Posts: 44639
Three circles with radii of 1, 2, and 3, lie on the same plane. Do any [#permalink]

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29 Jul 2015, 15:42
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67% (01:30) correct 33% (01:17) wrong based on 177 sessions

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Three circles with radii of 1, 2, and 3, lie on the same plane. Do any of these circles intersect or lie within another?

(1) If you connect the centers of the circles, an equilateral triangle with the height of 2√3 will be formed.
(2) The distance between the centers of any two circles is less than 6.

Kudos for a correct solution.
[Reveal] Spoiler: OA

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Manager
Joined: 20 Jul 2011
Posts: 80
GMAT 1: 660 Q49 V31
Re: Three circles with radii of 1, 2, and 3, lie on the same plane. Do any [#permalink]

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30 Jul 2015, 02:07
Bunuel wrote:
Three circles with radii of 1, 2, and 3, lie on the same plane. Do any of these circles intersect or lie within another?

(1) If you connect the centers of the circles, an equilateral triangle with the height of 2√3 will be formed.
(2) The distance between the centers of any two circles is less than 6.

Kudos for a correct solution.

St 1: An equilateral triangle with height 2$$\sqrt{3}$$ , then

the side of the triangle a = ( $$\sqrt{3}$$ / 2 ) * a = 2 $$\sqrt{3}$$

=> a = 4, which implies that circles with radii 3 and 2 must intersect each other to form a side of eq. triangle = 4.

Hence Sufficient.

St 2: Distance between any two centers < 6

if the distance between any two circles is less than 5, we can confirm that circles with radii 3 and 2 intersect each other.

But since the distance is < 6, then we cannot decide if they intersect each other. Hence not sufficient.

Option A
Manager
Joined: 14 Mar 2014
Posts: 148
GMAT 1: 710 Q50 V34
Re: Three circles with radii of 1, 2, and 3, lie on the same plane. Do any [#permalink]

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30 Jul 2015, 02:24
1
KUDOS
Bunuel wrote:
Three circles with radii of 1, 2, and 3, lie on the same plane. Do any of these circles intersect or lie within another?

(1) If you connect the centers of the circles, an equilateral triangle with the height of 2√3 will be formed.
(2) The distance between the centers of any two circles is less than 6.

Kudos for a correct solution.

IMO: A

Statement 1: If you connect the centers of the circles, an equilateral triangle with the height of 2√3 will be formed
Height of an equilateral triangle with side "a' = √3/2*a
2√3 = √3/2 *a
a= 4

If distance between any two centers is 4
For Circles with radius 1 and 2 --> do not intersect
For Circles with radius 1 and 3 --> Touch each other externally
For Circles with radius 2 and 3 --> intersect
Hence suff

Statement 2 : The distance between the centers of any two circles is less than 6.

Maximum distance between any two circles touching externally will be when circle with radius 2 and circle with radius 3 = 5
Thus If distance between them is
below 5 --> intersect
Equal to 5 --> touch externally
>5 --> do not intersect

Given condition distance < 6.
So we get different answers for d=5.5 or 5 or 4
Hence not suff
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Math Expert
Joined: 02 Sep 2009
Posts: 44639
Re: Three circles with radii of 1, 2, and 3, lie on the same plane. Do any [#permalink]

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17 Aug 2015, 09:10
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Expert's post
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This post was
BOOKMARKED
Bunuel wrote:
Three circles with radii of 1, 2, and 3, lie on the same plane. Do any of these circles intersect or lie within another?

(1) If you connect the centers of the circles, an equilateral triangle with the height of 2√3 will be formed.
(2) The distance between the centers of any two circles is less than 6.

Kudos for a correct solution.

800score Official Solution:

From Statement (1), we can determine how the circles are positioned relative to each other, because we know the distances between the centers. Therefore, we could establish whether any two of the circles intersect each other or one lies within another. Thus Statement (1) is sufficient. You do NOT need to do any further analysis of Statement (1).

Statement (2) is not sufficient. Imagine the two different situations: all the centers coincide; the centers are the vertices of an equilateral triangle with sides 5.5 . The first situation would answer the original question positively, the second one – negatively. Only if the distance was less than or equaled 5 could we guarantee the positive answer.

Remember that, in Data Sufficiency questions, it is not necessary to solve the problem, it is only necessary to establish whether or not we have sufficient information to do so.

Since Statement (1) is sufficient, and Statement (2) is not, the correct answer is A.
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Math Expert
Joined: 02 Sep 2009
Posts: 44639
Re: Three circles with radii of 1, 2, and 3, lie on the same plane. Do any [#permalink]

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17 Aug 2015, 09:21
Bunuel wrote:
Bunuel wrote:
Three circles with radii of 1, 2, and 3, lie on the same plane. Do any of these circles intersect or lie within another?

(1) If you connect the centers of the circles, an equilateral triangle with the height of 2√3 will be formed.
(2) The distance between the centers of any two circles is less than 6.

Kudos for a correct solution.

800score Official Solution:

From Statement (1), we can determine how the circles are positioned relative to each other, because we know the distances between the centers. Therefore, we could establish whether any two of the circles intersect each other or one lies within another. Thus Statement (1) is sufficient. You do NOT need to do any further analysis of Statement (1).

Statement (2) is not sufficient. Imagine the two different situations: all the centers coincide; the centers are the vertices of an equilateral triangle with sides 5.5 . The first situation would answer the original question positively, the second one – negatively. Only if the distance was less than or equaled 5 could we guarantee the positive answer.

Remember that, in Data Sufficiency questions, it is not necessary to solve the problem, it is only necessary to establish whether or not we have sufficient information to do so.

Since Statement (1) is sufficient, and Statement (2) is not, the correct answer is A.

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Hope it helps.
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Re: Three circles with radii of 1, 2, and 3, lie on the same plane. Do any [#permalink]

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12 Nov 2017, 20:11
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Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: Three circles with radii of 1, 2, and 3, lie on the same plane. Do any   [#permalink] 12 Nov 2017, 20:11
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