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Square Circles (Posted on 2004-05-27) Difficulty: 3 of 5
Given:

Three circles A, B and C.

Each circle is tangent to the other two.

The radius of A is 20.

The radius of B is 30.

Questions:

How many unique values of radius C exist where the centers of A, B and C form a right triangle? (Unique: Do not count triangles which are equal through flips and rotations. You may only count dissimilar triangles and similar triagles of differing sizes.)

What are the values?

See The Solution Submitted by Axorion    
Rating: 4.0000 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution the values | Comment 5 of 24 |

If A is internally tangent to B and the center of the small circle is at the right angle, the distance between their centers is 10, the longer leg is 20+r where r is the radius of C, and the hypotenuse is 30-r, so: 10^2+(20+r)^2 = (30-r)^2; this makes r = 4.

If the center of the large circle is the right angle, then 10^2+(30-r)^2 = (20+r)^2; thus r = 6.

If  A and B are externally tangent and both are external to C, A and B are separated by 50 units.

The center of C will then be at a right angle of the formed triangle if (20+r)^2 + (30+r)^2 = 50^2, so r = 10

The right angle will be at the center of A if (20+r)^2 + 50^2 = (30+r)^2; so r = 100

If A and B are externally tangent to each other but internal to C, then if the right angle is at the center of C then, (r-30)^2 +(r-20)^2 = 50^2; so r = 60

Have to go now... leave the last case till later.

 

 


  Posted by Charlie on 2004-05-27 16:01:05
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