2011 AMC 10A Problems/Problem 20
Contents
Problem 20
Two points on the circumference of a circle of radius are selected independently and at random. From each point a chord of length is drawn in a clockwise direction. What is the probability that the two chords intersect?
Solution 1
Fix a point from which we 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. (Imagine a regular hexagon inscribed in the circle)
Solution 2
Do what Solution 1 did until the guessing part. We then realize that the chords and radii make an equilateral triangle of length . Therefore the arc degree is The other arc degree is also Therefore the sum is Continue as follows.
See Also
2011 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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