If one side of a regular polygon with seven sides is produced, the exterior angle (in degrees) has the magnitude :

Options

A. `60`
B. `51 ""(3)/(7)`
C. `45`
D. `40`

Answer

To find the magnitude of the exterior angle when one side of a regular polygon with seven sides (heptagon) is produced, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the number of sides (n)**:
The polygon in question is a heptagon, which has 7 sides.
\[ n = 7 \]

**Hint**: Remember that a heptagon is defined as a polygon with 7 sides.

2. **Calculate the interior angle of the heptagon**:
The formula for the interior angle of a regular polygon is:
\[ \text{Interior Angle} = \frac{(n-2) \times 180}{n} \]
Substituting \( n = 7 \):
\[ \text{Interior Angle} = \frac{(7-2) \times 180}{7} = \frac{5 \times 180}{7} = \frac{900}{7} \approx 128.57^\circ \]

**Hint**: Use the formula for the interior angle, and remember to multiply by 180 degrees.

3. **Calculate the exterior angle**:
The exterior angle can be found using the relationship:
\[ \text{Exterior Angle} = 180^\circ - \text{Interior Angle} \]
Substituting the value we found:
\[ \text{Exterior Angle} = 180^\circ - \frac{900}{7} = \frac{1260}{7} - \frac{900}{7} = \frac{360}{7} \approx 51.43^\circ \]

**Hint**: The exterior angle is supplementary to the interior angle, so subtract the interior angle from 180 degrees.

4. **Determine the angle when one side is produced**:
When one side of the polygon is produced, the exterior angle remains the same. Thus, the magnitude of the exterior angle when one side is produced is:
\[ \text{Exterior Angle} = \frac{360}{7} \text{ degrees} \]

**Hint**: The exterior angle does not change when a side is produced; it remains the same.

### Final Answer:
The magnitude of the exterior angle when one side of a regular heptagon is produced is:
\[ \frac{360}{7} \text{ degrees} \approx 51.43^\circ \]