If each interior angle of a regular polygon is `7/6` times each angle of regular hexagon then what is the number of sides in the polygon

Options

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

Answer

To solve the problem, we need to find the number of sides (n) in a regular polygon given that each interior angle of the polygon is \( \frac{7}{6} \) times each angle of a regular hexagon.

### Step-by-step Solution:

1. **Identify the interior angle of a regular hexagon**:
A regular hexagon has 6 sides. The formula for the interior angle of a regular polygon with n sides is:
\[ \text{Interior angle} = \frac{(n-2) \times 180}{n} \]
For a hexagon (n = 6):
\[ \text{Interior angle of hexagon} = \frac{(6-2) \times 180}{6} = \frac{4 \times 180}{6} = 120^\circ \]

2. **Set up the equation for the polygon**:
According to the problem, the interior angle of the polygon is \( \frac{7}{6} \) times the interior angle of the hexagon:
\[ \text{Interior angle of polygon} = \frac{7}{6} \times 120^\circ = 140^\circ \]

3. **Set the interior angle formula for the polygon**:
Now, we can set the interior angle of the polygon equal to 140 degrees:
\[ \frac{(n-2) \times 180}{n} = 140 \]

4. **Cross-multiply to eliminate the fraction**:
Multiply both sides by n to get rid of the denominator:
\[ (n-2) \times 180 = 140n \]

5. **Expand and rearrange the equation**:
Expanding the left side gives:
\[ 180n - 360 = 140n \]
Now, rearranging the equation:
\[ 180n - 140n = 360 \]
\[ 40n = 360 \]

6. **Solve for n**:
Divide both sides by 40:
\[ n = \frac{360}{40} = 9 \]

Thus, the number of sides in the polygon is **9**.