If ABCDEF is a regular hexagon inscribed in a circle with centre O, then `overline(AB)+overline(AC)+overline(AD)+overline(AE)+overline(AF)=`

Options

A. `overline(AO)`
B. `5overline(AO)`
C. `6overline(AO)`
D. `8overline(AO)`

Answer

To solve the problem, we need to find the sum of the vectors \( \overline{AB} + \overline{AC} + \overline{AD} + \overline{AE} + \overline{AF} \) for a regular hexagon \( ABCDEF \) inscribed in a circle with center \( O \).

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- A regular hexagon has equal sides and equal angles. The vertices \( A, B, C, D, E, F \) are equally spaced around the circle.

2. **Vector Representation**:
- We can represent the vectors in terms of the position vectors from the center \( O \) to each vertex:
- Let \( \vec{A}, \vec{B}, \vec{C}, \vec{D}, \vec{E}, \vec{F} \) be the position vectors of points \( A, B, C, D, E, F \) respectively.

3. **Expressing the Required Sum**:
- We need to compute:
\[ \vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} \]
- Using the position vectors, we can express these as:
\[ \vec{AB} = \vec{B} - \vec{A}, \quad \vec{AC} = \vec{C} - \vec{A}, \quad \vec{AD} = \vec{D} - \vec{A}, \quad \vec{AE} = \vec{E} - \vec{A}, \quad \vec{AF} = \vec{F} - \vec{A} \]

4. **Substituting the Vectors**:
- Substitute the expressions into the sum:
\[ (\vec{B} - \vec{A}) + (\vec{C} - \vec{A}) + (\vec{D} - \vec{A}) + (\vec{E} - \vec{A}) + (\vec{F} - \vec{A}) \]
- This simplifies to:
\[ (\vec{B} + \vec{C} + \vec{D} + \vec{E} + \vec{F}) - 5\vec{A} \]

5. **Finding the Sum of the Position Vectors**:
- In a regular hexagon, the sum of the position vectors of the vertices is equal to \( 6\vec{O} \) (since the center \( O \) is the average of the vertices):
\[ \vec{B} + \vec{C} + \vec{D} + \vec{E} + \vec{F} = 5\vec{O} \]

6. **Substituting Back**:
- Substitute this result back into the sum:
\[ 5\vec{O} - 5\vec{A} = 5(\vec{O} - \vec{A}) \]

7. **Final Expression**:
- The expression \( \vec{O} - \vec{A} \) is a vector pointing from \( A \) to the center \( O \). Thus, we can express the final result as:
\[ 5(\vec{O} - \vec{A}) \]

### Conclusion:
The final result for the sum \( \overline{AB} + \overline{AC} + \overline{AD} + \overline{AE} + \overline{AF} \) is \( 5(\vec{O} - \vec{A}) \).