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

Question

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 .

VenoMfTw · Accepted Answer

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

ashokk138 · Answer

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

Bunuel · Answer

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 .

Bunuel · Answer

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

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