Difference between revisions of "2011 AMC 10A Problems/Problem 20"
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| + | == Problem 20 == | ||
| + | Two points on the circumference of a circle of radius r are selected independently and at random. From each point a chord of length r is drawn in a clockwise direction. What is the probability that the two chords intersect? | ||
| + | <math> \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2} </math> | ||
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| + | == Solution == | ||
| + | |||
| + | Fix a point <math>A</math> from which you draw a clockwise chord. In order for the clockwise chord from another point <math>B</math> to intersect that of point <math>A</math>, <math>A</math> and <math>B</math> must be no more than <math>r</math> units apart. By drawing the circle, we quickly see that <math>B</math> can be on <math>\frac{120}{360}=\boxed{\frac{1}{3} \ \textbf{(D)}}</math> of the perimeter of the circle. | ||
Revision as of 18:55, 11 February 2011
Problem 20
Two points on the circumference of a circle of radius r are selected independently and at random. From each point a chord of length r is drawn in a clockwise direction. What is the probability that the two chords intersect?
Solution
Fix a point 
from which you draw a clockwise chord. In order for the clockwise chord from another point 
to intersect that of point , and must be no more than 
units apart. By drawing the circle, we quickly see that can be on 
of the perimeter of the circle.
