Difference between revisions of "2007 AMC 10A Problems/Problem 24"
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<math>\text{(A)}\ \frac {8\sqrt {2}}{3} \qquad \text{(B)}\ 8\sqrt {2} - 4 - \pi \qquad \text{(C)}\ 4\sqrt {2} \qquad \text{(D)}\ 4\sqrt {2} + \frac {\pi}{8} \qquad \text{(E)}\ 8\sqrt {2} - 2 - \frac {\pi}{2}</math> | <math>\text{(A)}\ \frac {8\sqrt {2}}{3} \qquad \text{(B)}\ 8\sqrt {2} - 4 - \pi \qquad \text{(C)}\ 4\sqrt {2} \qquad \text{(D)}\ 4\sqrt {2} + \frac {\pi}{8} \qquad \text{(E)}\ 8\sqrt {2} - 2 - \frac {\pi}{2}</math> | ||
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The area we are trying to find is simply <math>ABFE-(\arc{AEC}+\triangle{ACO}+\triangle{BDO}+\arc{BFD}</math>. | The area we are trying to find is simply <math>ABFE-(\arc{AEC}+\triangle{ACO}+\triangle{BDO}+\arc{BFD}</math>. |
Revision as of 16:06, 27 April 2008
Problem
Circles centered at and
each have radius
, as shown. Point
is the midpoint of
, and
. Segments
and
are tangent to the circles centered at
and
, respectively, and
is a common tangent. What is the area of the shaded region
?
This problem needs a solution. If you have a solution for it, please help us out by adding it.
The area we are trying to find is simply $ABFE-(\arc{AEC}+\triangle{ACO}+\triangle{BDO}+\arc{BFD}$ (Error compiling LaTeX. Unknown error_msg).
Obviously, is parallel to
. Thus,
is a rectangle, and so its area is
.
Since
is tangent to $\circle{A}$ (Error compiling LaTeX. Unknown error_msg),
is a right
. We know
and
, so
is isosceles, a
-
right
, and has
with length
. The area of
. For obvious reasons, $\triangle{ACO}\congruent{\triangle{BDO}}$ (Error compiling LaTeX. Unknown error_msg), and so the area of
is also
.
$\arc{AEC}$ (Error compiling LaTeX. Unknown error_msg) (or $\arc{BFD}$ (Error compiling LaTeX. Unknown error_msg), for that matter) is
the area of its circle. Thus $\arc{AEC}$ (Error compiling LaTeX. Unknown error_msg) and $\arc{BFD}$ (Error compiling LaTeX. Unknown error_msg) both have an area of
.
Plugging all of these areas back into the original equation yields
.
See also
2007 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |