2017 AMC 12B Problems/Problem 18
Problem
The diameter of a circle of radius
is extended to a point
outside the circle so that
. Point
is chosen so that
and line
is perpendicular to line
. Segment
intersects the circle at a point
between
and
. What is the area of
?
Solution
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(8.865514650638614cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -6.36927122464312, xmax = 11.361758076634109, ymin = -3.789601803155515, ymax = 7.420015026296013; /* image dimensions */
draw((-2.,0.)--(0.6486486486486486,1.8918918918918919)--(2.,0.)--cycle);
/* draw figures */
draw(circle((0.,0.), 2.)); draw((-2.,0.)--(5.,5.)); draw((5.,5.)--(5.,0.)); draw((5.,0.)--(-2.,0.)); draw((-2.,0.)--(0.6486486486486486,1.8918918918918919)); draw((0.6486486486486486,1.8918918918918919)--(2.,0.)); draw((2.,0.)--(-2.,0.)); draw((2.,0.)--(5.,5.)); draw((0.,0.)--(5.,5.));
/* dots and labels */
dot((0.,0.),dotstyle);
label("", (-0.10330578512396349,-0.39365890308038826), NE * labelscalefactor);
dot((-2.,0.),dotstyle);
label("
", (-2.2370398196844437,-0.42371149511645134), NE * labelscalefactor);
dot((2.,0.),dotstyle);
label("
", (2.045454545454548,-0.36360631104432517), NE * labelscalefactor);
dot((5.,0.),dotstyle);
label("
", (4.900450788880542,-0.42371149511645134), NE * labelscalefactor);
dot((5.,5.),dotstyle);
label("
", (5.06574004507889,5.15104432757325), NE * labelscalefactor);
dot((0.6486486486486486,1.8918918918918919),linewidth(3.pt) + dotstyle);
label("
", (0.48271975957926694,2.100706235912847), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
Let be the center of the circle. Note that
. However, by Power of a Point,
, so
. Now
. Since
.
See Also
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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 12 Problems and Solutions |
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