The number of rectangles that can be obtained by joining four of the twelve vertices of a 12 - sided regular polygon is :

Options

A. 66
B. 30
C. 24
D. 15

Answer

To find the number of rectangles that can be formed by joining four of the twelve vertices of a 12-sided regular polygon, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Properties of Rectangles**:
- A rectangle is formed by selecting two pairs of opposite vertices. In a regular polygon, opposite vertices are those that are diametrically opposite to each other.

2. **Identifying the Pairs of Opposite Vertices**:
- In a 12-sided polygon, each vertex has one diametrically opposite vertex. For example, if we label the vertices as \( V_1, V_2, V_3, \ldots, V_{12} \), then the pairs of diametrically opposite vertices are:
- \( (V_1, V_7) \)
- \( (V_2, V_8) \)
- \( (V_3, V_9) \)
- \( (V_4, V_{10}) \)
- \( (V_5, V_{11}) \)
- \( (V_6, V_{12}) \)

3. **Counting the Pairs**:
- We have a total of 6 pairs of diametrically opposite vertices.

4. **Selecting Pairs to Form Rectangles**:
- To form a rectangle, we need to select 2 pairs from the 6 available pairs. The number of ways to choose 2 pairs from 6 pairs can be calculated using combinations:
\[ \text{Number of ways} = \binom{6}{2} \]

5. **Calculating the Combinations**:
- The formula for combinations is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
- Applying this to our case:
\[ \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]

6. **Conclusion**:
- Therefore, the total number of rectangles that can be formed by joining four of the twelve vertices of a 12-sided regular polygon is **15**.