2002 AMC 12B Problems/Problem 18
Problem
A point is randomly selected from the rectangular region with vertices . What is the probability that is closer to the origin than it is to the point ?
Solution
Solution 1
The region containing the points closer to than to is bounded by the perpendicular bisector of the segment with endpoints . The perpendicular bisector passes through midpoint of , which is , the center of the unit square with coordinates . Thus, it cuts the unit square into two equal halves of area . The total area of the rectangle is , so the area closer to the origin than to and in the rectangle is . The probability is .
Solution 2
Assume that the point is randomly chosen within the rectangle with vertices , , , . In this case, the region for to be closer to the origin than to point occupies exactly of the area of the rectangle, or square units.
If is chosen within the square with vertices , , , which has area square unit, it is for sure closer to .
Now if can only be chosen within the rectangle with vertices , , , , then the square region is removed and the area for to be closer to is then decreased by square unit, left with only square unit.
Thus the probability that is closer to is and that of is closer to the origin is .
~ Nafer
See also
2002 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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