Problem 25 · 1996 AJHSME
Stretch
Geometry & Measurement
geometric-probability
A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?
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Answer: A — 1/4.
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Hint 1 of 2
A point isn't 'in a region' — it just has two distances: how far from the center, and how far from the edge. If a point is r from the center of a radius-R circle, how far is it from the boundary? (Walk straight out to the rim.)
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Hint 2 of 2
Distance to the boundary is R − r. 'Closer to center' means r < R − r, which simplifies to r < R/2. So the winning points form a smaller circle of HALF the radius — now compare areas, not lengths.
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Approach: translate 'closer' into a smaller circle, then compare areas
- A point r from the center is R − r from the boundary (just continue straight out to the rim). 'Closer to the center' means r < R − r, i.e. r < R/2. So the favorable points fill the inner circle of radius R/2.
- Probability is the area ratio. Halving the radius quarters the area: (R/2)² ⁄ R² = 1/4.
- Watch the trap: halving the radius does NOT halve the area — area grows with radius SQUARED, so half the radius is a quarter of the area. That r² scaling is the heart of nearly every geometric-probability and similar-figures problem.
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