Final answer:
The probability that a randomly selected chord in a fixed circle is longer than the radius is 25%, as the chord must be within a smaller circle with radius half of the original.
Step-by-step explanation:
To calculate the probability that a randomly selected chord in a circle is longer than the radius, consider that for a chord to be longer than the radius of the circle, it must be closer to the center than the midpoint of the radius (this midpoint is half the radius from the center).
To find the probability that the length of a randomly selected chord on a fixed circle exceeds the radius, we need to determine the range of chord lengths that satisfy this condition.
Let's consider a circle with radius r. The length of a chord can be greater than the radius if it passes through the center of the circle. In this case, the length of the chord is equal to the diameter, which is 2 times the radius. Therefore, the probability that the length of a randomly selected chord exceeds the radius is 1/2 or 0.5.
The probability is thus equivalent to the chord being within a circle of radius ½ r centered at the original circle's center. Since the distance from the center for any chord is uniformly distributed from 0 to r, the probability that the chord is closer than ½ r from the center is the ratio of the area of the smaller circle (½ r) to the area of the larger circle (r), which is (½ r)^2 / r^2 = 1/4. Therefore, the probability is 1/4, or 25%.