+1641 Circles A, B, and C are externally tangent to each other. If AB = 14, BC = 18, and AC = 16, then find the radii of each of the circles.
+1490 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)
