A parallelogram has sides of 15 cm and 12 cm. If the distance between the 15 cm sides is 6 cm, find the distance between 12 cm sides.

Answer

To solve the problem step by step, we will find the distance between the 12 cm sides of the parallelogram.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- The lengths of the sides of the parallelogram are 15 cm and 12 cm.
- The distance (height) between the 15 cm sides is 6 cm.

2. **Understand the Area of a Parallelogram:**
- The area of a parallelogram can be calculated using the formula:
\[ \text{Area} = \text{Base} \times \text{Height} \]

3. **Calculate the Area Using the 15 cm Base:**
- Here, we take the base as 15 cm and the corresponding height as 6 cm.
- Therefore, the area (A) when using the 15 cm base is:
\[ A = 15 \, \text{cm} \times 6 \, \text{cm} = 90 \, \text{cm}^2 \]

4. **Calculate the Area Using the 12 cm Base:**
- Now, we take the base as 12 cm and let the height (the distance between the 12 cm sides) be \( h \) cm.
- The area in this case is:
\[ A = 12 \, \text{cm} \times h \, \text{cm} \]

5. **Set the Areas Equal:**
- Since the area of the parallelogram remains the same regardless of which base and height we use, we can set the two area expressions equal to each other:
\[ 15 \times 6 = 12 \times h \]
- This simplifies to:
\[ 90 = 12h \]

6. **Solve for \( h \):**
- To find \( h \), divide both sides by 12:
\[ h = \frac{90}{12} = 7.5 \, \text{cm} \]

7. **Conclusion:**
- The distance between the 12 cm sides of the parallelogram is 7.5 cm.