Regular polygons A and B have the number of sides in the ratio 1:2, and their interior angles are in the ratio 3:4. Then, the number of sides of polygon B equals:

Question

Accepted Answer

To solve this problem:

1. **Interior Angle Formula**:  
   The interior angle of an $$ n $$-sided regular polygon is:  
   $$
   	ext{Interior Angle} = \frac{(n - 2) 	imes 180}{n}.
   $$

2. **Given Ratios**:  
   - Number of sides of polygons A and B are in the ratio $$ 1:2 $$. Let the number of sides of polygon A be $$ n $$, so polygon B has $$ 2n $$ sides.  
   - Interior angles are in the ratio $$ 3:4 $$.

3. **Set Up Equation for Interior Angles**:  
   Using the formula for interior angles, equate their ratio:  
   $$
   \frac{\frac{(n - 2) 	imes 180}{n}}{\frac{(2n - 2) 	imes 180}{2n}} = \frac{3}{4}.
   $$

4. **Simplify**:  
   Cancel common terms and simplify:  
   $$
   \frac{(n - 2)}{n} \div \frac{(2n - 2)}{2n} = \frac{3}{4}.
   $$  
   $$
   \frac{(n - 2) \cdot 2n}{n \cdot (2n - 2)} = \frac{3}{4}.
   $$  
   $$
   \frac{2n(n - 2)}{n(2n - 2)} = \frac{3}{4}.
   $$

5. **Solve for $$ n $$**:  
   Simplify further:  
   $$
   \frac{2(n - 2)}{(2n - 2)} = \frac{3}{4}.
   $$  
   Cross-multiply:  
   $$
   8(n - 2) = 12(2n - 2).
   $$  
   Expand and simplify:  
   $$
   8n - 16 = 24n - 24.
   $$  
   $$
   16 = 16n.
   $$  
   $$
   n = 5.
   $$

6. **Find Number of Sides of Polygon B**:  
   Since polygon B has twice as many sides as polygon A:  
   $$
   	ext{Sides of Polygon B} = 2n = 2(5) = 10.
   $$

Thus, the number of sides of polygon B is **10**.