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 the line segments `A_0A_1, A_0A_2` and `A_0A_4` is

Options

A. `3//4`
B. `3sqrt3`
C. 3
D. `3sqrt3//2`

Explanation

To find the product of the lengths of the line segments \( A_0A_1 \), \( A_0A_2 \), and \( A_0A_4 \) in a regular hexagon inscribed in a circle of unit radius, we can follow these steps:

### Step 1: Understand the Geometry of the Hexagon
A regular hexagon has six equal sides and is symmetric. The vertices of the hexagon can be represented on the unit circle, where the center of the circle is at the origin \( O \) and the radius is 1.

### Step 2: Calculate Length \( A_0A_1 \)
The distance \( A_0A_1 \) is the length of one side of the hexagon. Since the radius of the circle is 1, and the angle subtended at the center \( O \) by the points \( A_0 \) and \( A_1 \) is \( \frac{2\pi}{6} = \frac{\pi}{3} \) radians (or 60 degrees), we can use the formula for the chord length:

\[ A_0A_1 = 2 \cdot r \cdot \sin\left(\frac{\theta}{2}\right) = 2 \cdot 1 \cdot \sin\left(\frac{\pi}{6}\right) = 2 \cdot 1 \cdot \frac{1}{2} = 1 \]

### Step 3: Calculate Length \( A_0A_2 \)
The distance \( A_0A_2 \) is the length of the diagonal skipping one vertex. The angle subtended at the center \( O \) by the points \( A_0 \) and \( A_2 \) is \( \frac{4\pi}{6} = \frac{2\pi}{3} \) radians (or 120 degrees). Using the chord length formula again:

\[ A_0A_2 = 2 \cdot r \cdot \sin\left(\frac{\theta}{2}\right) = 2 \cdot 1 \cdot \sin\left(\frac{2\pi}{6}\right) = 2 \cdot 1 \cdot \sin\left(\frac{\pi}{3}\right) = 2 \cdot 1 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \]

### Step 4: Calculate Length \( A_0A_4 \)
Similarly, the distance \( A_0A_4 \) is the length of the diagonal skipping two vertices. The angle subtended at the center \( O \) by the points \( A_0 \) and \( A_4 \) is \( \frac{6\pi}{6} = \pi \) radians (or 180 degrees). Thus:

\[ A_0A_4 = 2 \cdot r \cdot \sin\left(\frac{\theta}{2}\right) = 2 \cdot 1 \cdot \sin\left(\frac{\pi}{2}\right) = 2 \cdot 1 \cdot 1 = 2 \]

### Step 5: Calculate the Product
Now, we can calculate the product of the lengths:

\[ \text{Product} = A_0A_1 \cdot A_0A_2 \cdot A_0A_4 = 1 \cdot \sqrt{3} \cdot 2 = 2\sqrt{3} \]

### Final Answer
The product of the lengths \( A_0A_1 \), \( A_0A_2 \), and \( A_0A_4 \) is \( 2\sqrt{3} \).

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