Let `A_(0) A_(1)A_(2)A_(3)A_(4)A_(5)` be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments `A_(0) A_(1), A_(0)A_(2) and A_(0) A_(4)` is

Options

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

Answer

To find the product of the lengths of the line segments \( A_0 A_1 \), \( A_0 A_2 \), and \( A_0 A_4 \) in a regular hexagon inscribed in a circle of unit radius, we can follow these steps:

### Step 1: Identify the vertices of the hexagon
The vertices of the regular hexagon inscribed in a unit circle can be represented in the complex plane as:
- \( A_0 = e^{i \cdot 0} = 1 \)
- \( A_1 = e^{i \cdot \frac{\pi}{3}} = \frac{1}{2} + i \frac{\sqrt{3}}{2} \)
- \( A_2 = e^{i \cdot \frac{2\pi}{3}} = -\frac{1}{2} + i \frac{\sqrt{3}}{2} \)
- \( A_3 = e^{i \cdot \pi} = -1 \)
- \( A_4 = e^{i \cdot \frac{4\pi}{3}} = -\frac{1}{2} - i \frac{\sqrt{3}}{2} \)
- \( A_5 = e^{i \cdot \frac{5\pi}{3}} = \frac{1}{2} - i \frac{\sqrt{3}}{2} \)

### Step 2: Calculate the lengths of the segments
1. **Length of \( A_0 A_1 \)**:
\[ |A_0 A_1| = |1 - \left(\frac{1}{2} + i \frac{\sqrt{3}}{2}\right)| = |1 - \frac{1}{2} - i \frac{\sqrt{3}}{2}| = |\frac{1}{2} - i \frac{\sqrt{3}}{2}| \]
Using the formula for the modulus:
\[ |A_0 A_1| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

2. **Length of \( A_0 A_2 \)**:
\[ |A_0 A_2| = |1 - \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right)| = |1 + \frac{1}{2} - i \frac{\sqrt{3}}{2}| = |\frac{3}{2} - i \frac{\sqrt{3}}{2}| \]
Using the modulus:
\[ |A_0 A_2| = \sqrt{\left(\frac{3}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{3} = \sqrt{3} \]

3. **Length of \( A_0 A_4 \)**:
\[ |A_0 A_4| = |1 - \left(-\frac{1}{2} - i \frac{\sqrt{3}}{2}\right)| = |1 + \frac{1}{2} + i \frac{\sqrt{3}}{2}| = |\frac{3}{2} + i \frac{\sqrt{3}}{2}| \]
Using the modulus:
\[ |A_0 A_4| = \sqrt{\left(\frac{3}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{3} = \sqrt{3} \]

### Step 3: Calculate the product of the lengths
Now we can find the product of the lengths:
\[ |A_0 A_1| \cdot |A_0 A_2| \cdot |A_0 A_4| = 1 \cdot \sqrt{3} \cdot \sqrt{3} = 1 \cdot 3 = 3 \]

### Final Answer
The product of the lengths of the segments \( A_0 A_1 \), \( A_0 A_2 \), and \( A_0 A_4 \) is \( 3 \).

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