Another one with circles , distance , radii

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sapuna: Senior | Next Rank: 100 Posts; Posts: 67; Joined: Thu Jul 17, 2014 11:04 am; Thanked: 3 times; Followed by:2 members

Another one with circles , distance , radii

by sapuna » Sun Aug 03, 2014 12:43 am

Three circles with radii 1, 2 , 3, respectively, lie on the same plane. Do any of these circles intersect or lie completely within each other ?

1) IF you connect the centers of the circles, an equaliteral triangle is formed with a height of 2 square root of 3.

2) The distance between any two centers is less than 6.

Now, sorry for asking this, I thought I got the last question you guys explained and here I am again

From statement 1 we can find the side of the trianagle. What does this give us ?

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pm; Location: New Delhi, India; Thanked: 205 times; Followed by:24 members

by GMATinsight » Sun Aug 03, 2014 1:15 am

sapuna wrote:Three circles with radii 1, 2 , 3, respectively, lie on the same plane. Do any of these circles intersect or lie completely within each other ?

1) IF you connect the centers of the circles, an equaliteral triangle is formed with a height of 2 square root of 3.

2) The distance between any two centers is less than 6.

Now, sorry for asking this, I thought I got the last question you guys explained and here I am again From statement 1 we can find the side of the trianagle. What does this give us ?

Question : Three circles with radii 1, 2 , 3, respectively, lie on the same plane. Do any of these circles intersect or lie completely within each other ?

Statement 1) IF you connect the centers of the circles, an equaliteral triangle is formed with a height of 2 square root of 3.

Since (Sqrt3)xSide/2 = Height of Equilateral Triangle

therefore, (Sqrt3)xSide/2 = 2(Sqrt3)

i.e. Side of equilateral Triangle = 4

From Statement 1 we know that side of the Equilateral triangle is 4
Side of Traingle 4 (Distance between centers) is possible only when two Circles with radii 3 and 1 are Tangant

As well as when two Circles with radii 3 and 2 are intersecting at two points
SUFFICIENT

Statement 2) The distance between any two centers is less than 6.
INSUFFICIENT

Answer: Option A

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sapuna: Senior | Next Rank: 100 Posts; Posts: 67; Joined: Thu Jul 17, 2014 11:04 am; Thanked: 3 times; Followed by:2 members

by sapuna » Sun Aug 03, 2014 1:58 am

Ah , you misunderstood what I was asking I think. I already knew the side of the triangle was 4. I was asking why is this true :

Side of Traingle 4 (Distance between centers) is possible only when two Circles with radii 3 and 1 are Tangant

As well as when two Circles with radii 3 and 2 are intersecting at two points
SUFFICIENT

Statement 2) The distance between any two centers is less than 6 - in the answers I have it`s said that the distance has to be less than 5 for the statement to hold

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pm; Location: New Delhi, India; Thanked: 205 times; Followed by:24 members

by GMATinsight » Sun Aug 03, 2014 3:26 am

sapuna wrote:Ah , you misunderstood what I was asking I think. I already knew the side of the triangle was 4. I was asking why is this true :

Side of Traingle 4 (Distance between centers) is possible only when two Circles with radii 3 and 1 are Tangant

As well as when two Circles with radii 3 and 2 are intersecting at two points
SUFFICIENT

Statement 2) The distance between any two centers is less than 6 - in the answers I have it`s said that the distance has to be less than 5 for the statement to hold

We have to check this first

R1 + R2 = 1+2 = 3
R2 + R3 = 2+3 = 5
R1 + R3 = 1+3 = 4

i.e. If Circles with R1 and R2 touch each other at just 1 point (i.e. they are Tangent) then the distance between their centers will be 3 units

i.e. the increase in distance between their centers will cause them to go away and they will not touch.

For two circles to intersect (or move completely in the other circles) the distance between their centers must lesser than or equal to Sum of their Radii
Distance between centers < R1+R2 [The figure is as attached]

Since the maximum sum of radii of any two circles out of given is 5 therefore if the equilateral triangle is drawn by joining their centers then the three will NOT intersect only if the length of the side of Equilateral Triangle is greater than 5 units

Attachments

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Sun Aug 03, 2014 3:41 am

sapuna wrote:Three circles with radii 1, 2 , 3, respectively, lie on the same plane. Do any of these circles intersect or lie completely within each other ?

1) IF you connect the centers of the circles, an equaliteral triangle is formed with a height of 2 square root of 3.

2) The distance between any two centers is less than 6.

Figure 1:

In the figure above, x, y and z represent the distances between circles A, B and C.
âˆ†ABC is formed by joining the 3 centers.

Statement 1: If you connect the centers of the circles, an equaliteral triangle is formed with a height of 2 square root of 3.
Figure 2:

Thus, each side of âˆ†ABC = 4.
In figure 1, BC=4 only if z<0, implying that circles B and C intersect.
SUFFICIENT.

Statement 2: The distance between any two centers is less than 6.
It's possible that x=y=z=0, in which all 3 circles intersect.
It's possible that x=y=z=1/2, in which case none of the circles intersect.
INSUFFICIENT.

The correct answer is A.

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