Let `A_0A_1A_2A_3A_4A_5` be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments `A_0A_1,A_0A_2,A_0A_4` is

Options

A. `3/4`
B. `root(3)(3)`
C. 3
D. `root(3)(3)/2`

Answer

To solve the problem, we need to find the product of the lengths of the segments \( A_0A_1 \), \( A_0A_2 \), and \( A_0A_4 \) in a regular hexagon inscribed in a circle of unit radius.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- A regular hexagon inscribed in a circle means all vertices of the hexagon lie on the circumference of the circle.
- The radius of the circle is given as 1.

2. **Coordinates of the Vertices**:
- The vertices of the hexagon can be represented in the coordinate system using angles. The vertices \( A_0, A_1, A_2, A_3, A_4, A_5 \) can be defined as:
- \( A_0 = (1, 0) \)
- \( A_1 = \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \)
- \( A_2 = \left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right) \)
- \( A_3 = (-1, 0) \)
- \( A_4 = \left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right) \)
- \( A_5 = \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) \)

3. **Calculating the Lengths of the Segments**:
- **Length \( A_0A_1 \)**:
\[ A_0A_1 = \sqrt{\left(1 - \frac{1}{2}\right)^2 + \left(0 - \frac{\sqrt{3}}{2}\right)^2} = \sqrt{\left(\frac{1}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

- **Length \( A_0A_2 \)**:
\[ A_0A_2 = \sqrt{\left(1 - \left(-\frac{1}{2}\right)\right)^2 + \left(0 - \frac{\sqrt{3}}{2}\right)^2} = \sqrt{\left(1 + \frac{1}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\left(\frac{3}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2} \]
\[ = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{3} = \sqrt{3} \]

- **Length \( A_0A_4 \)**:
\[ A_0A_4 = \sqrt{\left(1 - \left(-\frac{1}{2}\right)\right)^2 + \left(0 + \frac{\sqrt{3}}{2}\right)^2} = \sqrt{\left(1 + \frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\left(\frac{3}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} \]
\[ = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{3} \]

4. **Calculating the Product**:
- Now, we multiply the lengths obtained:
\[ A_0A_1 \times A_0A_2 \times A_0A_4 = 1 \times \sqrt{3} \times \sqrt{3} = 1 \times 3 = 3 \]

### Final Answer:
The product of the lengths of the line segments \( A_0A_1, A_0A_2, A_0A_4 \) is \( 3 \).

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