MathDB

Circles with centers

O

and

P

have radii 2 and 4, respectively, and are externally tangent. Points

A

and

B

are on the circle centered at

O

, and points

C

and

D

are on the circle centered at

P

, such that

\overline{AD}

and

\overline{BC}

are common external tangents to the circles. What is the area of hexagon

AOBCPD

? [asy] size(250);defaultpen(linewidth(0.8)); pair X=(-6,0), O=origin, P=(6,0), B=tangent(X, O, 2, 1), A=tangent(X, O, 2, 2), C=tangent(X, P, 4, 1), D=tangent(X, P, 4, 2); pair top=X+15*dir(X--A), bottom=X+15*dir(X--B); draw(Circle(O, 2)^^Circle(P, 4)); draw(bottom--X--top); draw(A--O--B^^O--P^^D--P--C); pair point=X; label("

2

", midpoint(O--A), dir(point--midpoint(O--A))); label("

4

", midpoint(P--D), dir(point--midpoint(P--D))); label("

O

", O, SE); label("

P

", P, dir(point--P)); pair point=O; label("

A

", A, dir(point--A)); label("

B

", B, dir(point--B)); pair point=P; label("

C

", C, dir(point--C)); label("

D

", D, dir(point--D)); fill((-3,7)--(-3,-7)--(-7,-7)--(-7,7)--cycle, white);[/asy]

<span class='latex-bold'>(A) </span> 18\sqrt {3} \qquad <span class='latex-bold'>(B) </span> 24\sqrt {2} \qquad <span class='latex-bold'>(C) </span> 36 \qquad <span class='latex-bold'>(D) </span> 24\sqrt {3} \qquad <span class='latex-bold'>(E) </span> 32\sqrt {2}

Problem Statement