The ratio between the number of sides of two regular Polygons is 1:2 and ratio between their interior angles is 2 : 3, The number of sides of these polygons are respectively

Options

A. 7,4
B. 6,12
C. 5,10
D. 4,8

Answer

To solve the problem, we need to find the number of sides of two regular polygons given their ratios of sides and interior angles.

### Step-by-Step Solution:

1. **Define Variables**:
Let \( n_1 \) be the number of sides of the first polygon and \( n_2 \) be the number of sides of the second polygon. According to the problem, the ratio of the number of sides is given as:
\[ \frac{n_1}{n_2} = \frac{1}{2} \]
This means:
\[ n_2 = 2n_1 \]

2. **Interior Angle Formula**:
The formula for the interior angle of a regular polygon with \( n \) sides is:
\[ \text{Interior Angle} = \frac{(n-2) \times 180}{n} \]
Thus, for the first polygon:
\[ \text{Interior Angle}_1 = \frac{(n_1 - 2) \times 180}{n_1} \]
And for the second polygon:
\[ \text{Interior Angle}_2 = \frac{(n_2 - 2) \times 180}{n_2} \]

3. **Set Up the Ratio of Interior Angles**:
We know the ratio of the interior angles is given as:
\[ \frac{\text{Interior Angle}_1}{\text{Interior Angle}_2} = \frac{2}{3} \]
Substituting the expressions for the interior angles:
\[ \frac{\frac{(n_1 - 2) \times 180}{n_1}}{\frac{(n_2 - 2) \times 180}{n_2}} = \frac{2}{3} \]
This simplifies to:
\[ \frac{(n_1 - 2) \times n_2}{(n_2 - 2) \times n_1} = \frac{2}{3} \]

4. **Substituting \( n_2 \)**:
Substitute \( n_2 = 2n_1 \) into the equation:
\[ \frac{(n_1 - 2) \times (2n_1)}{(2n_1 - 2) \times n_1} = \frac{2}{3} \]
Simplifying the left side:
\[ \frac{2(n_1 - 2)}{2 - \frac{2}{n_1}} = \frac{2}{3} \]
This simplifies to:
\[ \frac{(n_1 - 2)}{(n_1 - 1)} = \frac{2}{3} \]

5. **Cross-Multiplying**:
Cross-multiply to solve for \( n_1 \):
\[ 3(n_1 - 2) = 2(n_1 - 1) \]
Expanding both sides:
\[ 3n_1 - 6 = 2n_1 - 2 \]
Rearranging gives:
\[ 3n_1 - 2n_1 = 4 \implies n_1 = 4 \]

6. **Finding \( n_2 \)**:
Now substitute \( n_1 \) back to find \( n_2 \):
\[ n_2 = 2n_1 = 2 \times 4 = 8 \]

### Final Answer:
The number of sides of the two polygons are \( n_1 = 4 \) and \( n_2 = 8 \).