1976 AHSME Problems/Problem 8

Problem

A point in the plane, both of whose rectangular coordinates are integers with absolute values less than or equal to four, is chosen at random, with all such points having an equal probability of being chosen. What is the probability that the distance from the point to the origin is at most two units?

$\textbf{(A) }\frac{13}{81}\qquad \textbf{(B) }\frac{15}{81}\qquad \textbf{(C) }\frac{13}{64}\qquad \textbf{(D) }\frac{\pi}{16}\qquad \textbf{(E) }\text{the square of a rational number}$

Solution 1

The rectangular region will be an 8x8 grid which is centered at the origin. Since the distance from the point chosen to the origin is at most two units, the favorable region will NOT be the $2 \times 2$ grid centered at the origin. It will be a circle of radius two units centered at the origin. The area of that circle is pi, and the area of the total region is 64. Therefore, the answer is \boxed{\textbf{(D) }\frac{\pi}{16}}

1976 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
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1976 AHSME Problems/Problem 8

Problem

Solution 1

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