M14-02

Question

If three circles having radii 1, 2, and 3 respectively lie on a plane, do any two of these circles intersect?

(1) Centers of the circles form an equilateral triangle with height  2\sqrt{3} .

(2) Centers of the circles do not lie on the same line.

Bunuel · Answer

Official Solution:

Statement (1) by itself is sufficient. From S1 we know how the circles are positioned relative to each other (we know the distances between the centers). Therefore, we can answer the question whether the circles cross or not.
 
Statement (1) by itself is insufficient. The information that is missing is the distance between the centers.

Answer: A

Richa07 · Answer

why statement is insufficent

bhatiavai · Answer

What does the second statement imply ?

One can always connect the centers of two circles by a line ?

Bunuel · Answer

We have 3 circles not 2. 3 points are not necessarily on one line.

Icecream87 · Answer

Hi Bunuel, 
Could you please enlighten me on this:
I don't get the answer to this question. How do we know whether the circles cross or not with just the height of the triangle? And with no info on the circles' radii? 
Statement (1) by itself is sufficient. From S1 we know how the circles are positioned relative to each other (we know the distances between the centers).   Therefore, we can answer the question whether the circles cross or not.   
Does this mean they cross or they don't?

And what does statement two mean?

And is my diagram below ok?
Thanks alot

Bunuel · Answer

Please check the discussion here: if-three-circles-having-radii-1-2-and-3-respectively-lie-o-54757.html

Icecream87 · Answer

Got it, thank you Bunuel!

sukoon9334 · Answer

Icecream87 
Explaining Statement (1) Sufficiency:

We've been able to calculate the distance using the short-cuts of the equilateral triangle. 
In an equilateral triangle of side 'a', the height is  \sqrt{3}a/2 . 
The given height is 2√3. 
Solving for 'a', we get a= 4. 
Thus, we know that a circle of radii 2 and 3 will definitely intersect.

Zake · Answer

If we want to calculate statement 1 you can do so the following way.

The height of an equilateral triangle creates 2 right angle triangles with 30-60-90 degrees. Remember, 30-60-90 triangle has the following concept on the length of each side x, x√3, and 2x. This implies that the height is the length opposite the 60 degree angle. This tells us that what is opposite the 90 degree angle is also the side length of the equilateral triangle.

Opposite 90 degree angle is 2x. -> 2(2) -> 4.

If radius equal 1,2,3, then this means that the diameter equals 2,4,6 (double radius). This tells us that the circles do touch.

M14-02

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