2004 AMC 12A Problem 19

Question

Circles

A,

B,

and

C

are externally tangent to each other and internally tangent to circle

D.

Circles

B

and

C

are congruent. Circle

A

has radius

1

and passes through the center of

D.

What is the radius of circle

B?

\dfrac{2}{3}

\dfrac{\sqrt{3}}{2}

\dfrac{7}{8}

\dfrac{8}{9}

\dfrac{1 + \sqrt{3}}{3}

Answer: D

Concepts:tangent circles Pythagorean Theorem system of equations

Difficulty rating: 1920

Small hint:

Since

A

passes through the center of

D

and is internally tangent, circle

D

has radius

2

Big hint:

Place the center of

D

at the origin; use right triangles formed by the centers and the axis of symmetry

Solution:

Circle

A

has radius

1

and passes through the center of

D

while being internally tangent to

D,

so

D

has radius

2.

Place the center of

D

at the origin, with

A

centered at

(-1, 0).

By symmetry,

B

has center

(x, y)

and radius

r,

with

C

its mirror image across the horizontal axis, so the two congruent circles touch on that axis and

y = r.

Internal tangency to

D

gives

x^2 + y^2 = (2 - r)^2,

and external tangency to

A

gives

(x + 1)^2 + y^2 = (1 + r)^2.

Subtracting and using

y = r

yields

x = \tfrac23

and

r = \tfrac89.

The radius of circle

B

is

\tfrac89.

Thus, the correct answer is D.

2004 AMC 12A Problem 19

Answer: D

Difficulty rating: 1920

Small hint:

Big hint:

Solution:

Problem 19 in other years