If two points are taken at random on the circumference of circle, the chance that their distance apart is greater than the radius of the circle is

Options

A. `1/4`
B. `1/3`
C. `2/3`
D. `3/4`

Answer

To solve the problem of finding the probability that the distance between two randomly chosen points on the circumference of a circle is greater than the radius of the circle, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Circle**:
- Consider a circle with center O and radius r. We need to select two points A and B on the circumference of the circle.

2. **Distance Between Points**:
- The distance between points A and B can be represented as the chord length of the circle. The distance between two points on the circumference is greater than the radius if the angle subtended by the chord at the center of the circle is greater than 60 degrees.

3. **Angle Subtended**:
- The maximum angle that can be subtended by the points A and B at the center O is 360 degrees. If the angle ∠AOB is greater than 60 degrees, then the distance AB will be greater than the radius r.

4. **Calculating Favorable Outcomes**:
- The total angle for which the distance AB is greater than r is from 60 degrees to 300 degrees (360 degrees - 60 degrees). This gives us a range of 240 degrees.

5. **Calculating Total Outcomes**:
- The total possible angle is 360 degrees.

6. **Finding Probability**:
- The probability that the distance between points A and B is greater than the radius is given by the ratio of the favorable outcomes to the total outcomes:
\[ P(\text{Distance } > r) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{240}{360} = \frac{2}{3} \]

### Final Answer:
The probability that the distance between two randomly chosen points on the circumference of the circle is greater than the radius of the circle is \( \frac{2}{3} \).