A point is selected at random from inside a circle. The probability that the point is closer to the circumference of the circle than to its centre, is

Options

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

Answer

To solve the problem of finding the probability that a point selected at random from inside a circle is closer to the circumference than to its center, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to determine the area inside the circle where points are closer to the circumference than to the center.

2. **Draw the Circle**: Let’s consider a circle with radius \( r \). The center of the circle is denoted as point \( O \).

3. **Identify the Condition**: A point \( P \) inside the circle is closer to the circumference than to the center if the distance from \( P \) to the circumference is less than the distance from \( P \) to the center \( O \).

4. **Set up the Relationship**: The distance from point \( P \) to the center \( O \) is \( OP \), and the distance from \( P \) to the circumference is \( r - OP \). We need to find when:
\[ r - OP < OP \]
Simplifying this gives:
\[ r < 2 \cdot OP \quad \Rightarrow \quad OP > \frac{r}{2} \]

5. **Define the Inner Circle**: The points that are closer to the center than to the circumference are those within a smaller circle of radius \( \frac{r}{2} \) centered at \( O \).

6. **Calculate Areas**:
- **Total Area of the Larger Circle**:
\[ A_{\text{total}} = \pi r^2 \]
- **Area of the Smaller Circle** (where points are closer to the center):
\[ A_{\text{inner}} = \pi \left(\frac{r}{2}\right)^2 = \pi \frac{r^2}{4} \]

7. **Area Where Points are Closer to the Circumference**:
\[ A_{\text{event}} = A_{\text{total}} - A_{\text{inner}} = \pi r^2 - \pi \frac{r^2}{4} = \pi r^2 \left(1 - \frac{1}{4}\right) = \pi r^2 \cdot \frac{3}{4} = \frac{3\pi r^2}{4} \]

8. **Calculate the Probability**:
The probability \( P \) that a randomly selected point is closer to the circumference than to the center is given by the ratio of the area where this condition holds to the total area:
\[ P = \frac{A_{\text{event}}}{A_{\text{total}}} = \frac{\frac{3\pi r^2}{4}}{\pi r^2} = \frac{3}{4} \]

### Final Answer:
The probability that a point selected at random from inside the circle is closer to the circumference than to its center is \( \frac{3}{4} \).

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