An exterior angle and an interior angle of a regular polygon are in the ratio `2:7`. Find the number of sides in the polygon.

Options

A. `8`
B. `7`
C. `10`
D. `9`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Problem**: We are given that the ratio of an exterior angle to an interior angle of a regular polygon is \(2:7\). We need to find the number of sides (n) of the polygon.

2. **Formulas for Angles**:
- The formula for the interior angle of a regular polygon is:
\[ \text{Interior Angle} = \frac{180(n - 2)}{n} \]
- The formula for the exterior angle of a regular polygon is:
\[ \text{Exterior Angle} = \frac{360}{n} \]

3. **Setting Up the Ratio**: According to the problem, the ratio of the exterior angle to the interior angle is:
\[ \frac{\text{Exterior Angle}}{\text{Interior Angle}} = \frac{2}{7} \]
Substituting the formulas, we have:
\[ \frac{\frac{360}{n}}{\frac{180(n - 2)}{n}} = \frac{2}{7} \]

4. **Simplifying the Ratio**: We can simplify the left side:
\[ \frac{360}{180(n - 2)} = \frac{2}{7} \]
This simplifies to:
\[ \frac{2}{n - 2} = \frac{2}{7} \]

5. **Cross Multiplying**: To eliminate the fractions, we can cross-multiply:
\[ 2 \cdot 7 = 2(n - 2) \]
This simplifies to:
\[ 14 = 2(n - 2) \]

6. **Solving for n**: Now, we can solve for \(n\):
\[ 14 = 2n - 4 \]
Adding 4 to both sides gives:
\[ 18 = 2n \]
Dividing both sides by 2 gives:
\[ n = 9 \]

7. **Conclusion**: The number of sides in the regular polygon is \(n = 9\).