Find the radii of the circles A, B, and C

Here's a very clever method to solve this problem:

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Let a, b, and c be the radii of circles A, B, and C respectively. From the, Segment Addition Postulate we know that:

 

AB = a + b = 14

BC = b + c = 18

AC = a + c = 16

 

We have three equations and three unknowns:

 

a + b = 14

b + c = 18

a + c = 16

 

Let's eliminate the a by multiplying the first equation by -1 then adding them together: I'll line up the a's b's and c's, putting in a 0 if one of them is missing:

 

-a - b + 0 = -14

0 + b + c = 18

a + 0 + c = 16

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0 + 0 + 2c = 20

 

2c = 20

c = 10 The radius of circle C is 10.

 

To find b:

 

b + c = 18

b + 10 = 18

b = 8 The radius of Circle B is 8

 

To find a, use a + b = 14, knowing that b = 8.

 

(Solved by Philip P)