Circles with centers at O and P have radii 2 and 4,
respectively, and are externally tangent. Points A and
B on the circle with center O and points C and D on
the circle with center P are such that AD and BC are
common external tangents to the circles. What is the
area of the concave hexagon AOBCP D ?
√
(A) 18 3
√
(B) 24 2
√
(D) 24 3
(C) 36
D
A
4
2
6
B
√
(E) 32 2
2006 AMC 10 B, Problem #24—
2006 AMC 12 B, Problem #15—
“Break up the region into 2 rectangles and 2 triangles.”
Solution (B) Through O draw a line parallel to AD intersecting P D at F .
D
2
A
2
F
2
6
P
O
B
C
Then AOF D is a rectangle
and OP F is a right triangle. √
Thus DF = 2,
√
F P = 2, and OF = 4 2. The √
area of trapezoid
AOP
D
is
12
2, and the area
√
of hexagon AOBCP D is 2 · 12 2 = 24 2.
OR
Lines AD, BC, and OP intersect at a common point H.
D
A
4
2
6
H
O
P
B
C
Because ∠P DH = ∠OAH = 90◦ , triangles P DH and OAH are similar with
ratio of √
similarity 2. Thus 2HO
√ = HP = HO+OP = HO+6, so HO =√6 and
2
2
AH
√ = HO − OA = 4 2. Hence the
√ area of√4OAH is (1/2)(2)(4 2) =
4 2, and the area of 4P DH is (22 )(4 2) = 16 2. The area of the hexagon
√
is twice the area of 4P DH minus twice the area of 4OAH, so it is 24 2.
Difficulty: Hard
Difficulty: Medium-hard
NCTM Standard: Geometry Standard: analyze characteristics and properties of two- and threedimensional geometric shapes and develop mathematical arguments about geometric relationships
Mathworld.com Classification: Geometry > Line Geometry > Concurrence > Tangent
Externally
P
O
C