Difference between revisions of "2010 AMC 10B Problems/Problem 19"
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Revision as of 12:02, 4 July 2013
Problem
A circle with center has area
. Triangle
is equilateral, $\overbar{BC}$ (Error compiling LaTeX. Unknown error_msg) is a chord on the circle,
, and point
is outside
. What is the side length of
?
Solution
The formula for the area of a circle is so the radius of this circle is
Because must be in the interior of circle
![[asy] unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(11pt)); dotfactor=3; real r=sqrt(156); pair A=(0,sqrt(48)), B=(-3,sqrt(147)), C=(3,sqrt(147)); pair O=(0,0); pair X=(0,7sqrt(3)); path outer=Circle(O,r); draw(outer); draw(A--B--C--cycle); draw(O--X); draw(O--B); pair[] ps={A,B,C,O,X}; dot(ps); label("$A$",A,SE); label("$B$",B,NW); label("$C$",C,NE); label("$O$",O,S); label("$X$",X,N); label("$s$",A--C,SE); label("$\frac{s}{2}$",B--X,N); label("$\frac{s\sqrt{3}}{2}$",A--X,NE); label("$\sqrt{156}$",O--B,SW); label("$4\sqrt{3}$",A--O,E); [/asy]](http://latex.artofproblemsolving.com/0/3/f/03fe10542785e49de90d325a8bb86d13462ffd6a.png)
Let be the unknown value, the sidelength of the triangle, and let
be the point on
where
Since
is equilateral,
and
We are given
Use the Pythagorean Theorem and solve for
See Also
2010 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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