The ratio of sides of two regular polygons is 1 : 2 and the ratio of their interior angles is 2 : 3. The sides of regular polygons are :

Question

Accepted Answer

To determine why Option 1 is the correct answer, we need to analyze the given conditions about the regular polygons and their properties.

### Explanation for Option 1 (4, 8):

1. **Understanding Regular Polygons:**
   - A regular polygon is a polygon with all sides and angles equal.
   - The formula for the interior angle of a regular polygon with $ n $ sides is given by:
     $$
     \text{Interior Angle} = \frac{(n-2) \times 180}{n}
     $$

2. **Given Ratios:**
   - The ratio of the number of sides of the two polygons is 1:2.
   - The ratio of their interior angles is 2:3.

3. **Applying the Ratios:**
   - Let the number of sides of the first polygon be $ n $. Then, the number of sides of the second polygon is $ 2n $.
   - The interior angle of the first polygon is:
     $$
     \frac{(n-2) \times 180}{n}
     $$
   - The interior angle of the second polygon is:
     $$
     \frac{(2n-2) \times 180}{2n}
     $$

4. **Setting Up the Ratio of Interior Angles:**
   - According to the problem, the ratio of the interior angles is 2:3. Therefore:
     $$
     \frac{\frac{(n-2) \times 180}{n}}{\frac{(2n-2) \times 180}{2n}} = \frac{2}{3}
     $$

5. **Solving the Equation:**
   - Simplifying the equation:
     $$
     \frac{(n-2) \times 2n}{(2n-2) \times n} = \frac{2}{3}
     $$
   - Cross-multiplying gives:
     $$
     3(n-2) \times 2n = 2(2n-2) \times n
     $$
   - Simplifying further:
     $$
     6n^2 - 12n = 4n^2 - 4n
     $$
     $$
     2n^2 - 8n = 0
     $$
     $$
     2n(n - 4) = 0
     $$
   - Solving for $ n $, we get $ n = 4 $.

6. **Conclusion:**
   - The first polygon has 4 sides, and the second polygon has $ 2 \times 4 = 8 $ sides.
   - Therefore, the sides of the regular polygons are 4 and 8, which matches Option 1.

### Explanation for Why Other Options are Incorrect:

- **Option 2 (5, 10):**
  - If the first polygon had 5 sides, the second would have 10 sides. The interior angle ratio would not satisfy the given 2:3 ratio when calculated using the formula for interior angles.

- **Option 3 (6, 12):**
  - Similarly, with 6 and 12 sides, the interior angle ratio would not match the required 2:3 ratio.

- **Option 4 (8, 16):**
  - For 8 and 16 sides, the interior angle ratio would again not satisfy the condition of being 2:3.

Thus, Option 1 is the only set of side numbers that satisfies both the given side ratio and the interior angle ratio, confirming it as the correct answer.