Difference between revisions of "2006 AMC 10B Problems/Problem 19"
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Revision as of 12:18, 4 July 2013
Problem
A circle of radius is centered at
. Square
has side length
. Sides
and
are extended past
to meet the circle at
and
, respectively. What is the area of the shaded region in the figure, which is bounded by
,
, and the minor arc connecting
and
?
Solution
The shaded area is equivalent to the area of sector , minus the area of triangle
plus the area of triangle
.
Using the Pythagorean Theorem, so
.
Clearly, and
are
triangles with
. Since
is a square,
.
can be found by doing some subtraction of angles.
So, the area of sector is
.
The area of triangle is
.
Since ,
. So, the area of triangle
is
. Therefore, the shaded area is
See Also
2006 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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 |
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