2007 AMC 10A Problems/Problem 24
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 
?
Solution
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, 
. Thus, 
is a rectangle, and so its area is 
.
Since 
is tangent to circle , 
is a right 
. We know 
and 
, so is isosceles, a 
- right , and has 
with length . The area of 
. By symmetry, 
, 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
