1983 AMC 12 Problem 23

Question

In the adjoining figure the five circles are tangent to one another consecutively and to the lines

L_1

and

L_2.

If the radius of the largest circle is

18

and that of the smallest one is

8,

then the radius of the middle circle is

12

12.5

13

13.5

14

Answer: A

Concepts:tangent circles similarity geometric sequence

Difficulty rating: 2310

Small hint:

Every pair of consecutive tangent circles has the same shape after scaling

Big hint:

The radii form a geometric sequence, so the middle radius squared is the product of the extremes

Solution:

The centers lie on the angle bisector of

L_1

and

L_2.

Similarity of the configuration for any two consecutive circles shows that consecutive radii have a constant ratio. Thus the five radii form a geometric sequence. If the middle radius is

m,

symmetry of a five-term geometric sequence gives

m^2=8\cdot18,

so

m=12.

Therefore, the correct answer is A.