The adjacent sides of a parallelogram are 12 cm and 10 cm, respectively. If the perpendicular distance between the longer sides is 5 cm, find the perpendicular distance between the shorter sides.

Answer

To solve the problem, we need to find the perpendicular distance between the shorter sides of the parallelogram. Here’s the step-by-step solution:

### Step 1: Identify the given values
We have a parallelogram with:
- Length of the longer side (base) = 12 cm
- Length of the shorter side (base) = 10 cm
- Perpendicular distance between the longer sides (height) = 5 cm

### Step 2: Calculate the area using the longer side
The area of a parallelogram can be calculated using the formula:
\[ \text{Area} = \text{Base} \times \text{Height} \]

Using the longer side (12 cm) as the base and the given height (5 cm):
\[ \text{Area} = 12 \, \text{cm} \times 5 \, \text{cm} = 60 \, \text{cm}^2 \]

### Step 3: Set up the equation for the area using the shorter side
Now, we will use the shorter side (10 cm) as the base and let the perpendicular distance (height) between the shorter sides be \( x \):
\[ \text{Area} = 10 \, \text{cm} \times x \]

### Step 4: Equate the two area expressions
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:
\[ 60 \, \text{cm}^2 = 10 \, \text{cm} \times x \]

### Step 5: Solve for \( x \)
To find \( x \), divide both sides of the equation by 10 cm:
\[ x = \frac{60 \, \text{cm}^2}{10 \, \text{cm}} \]
\[ x = 6 \, \text{cm} \]

### Conclusion
The perpendicular distance between the shorter sides of the parallelogram is **6 cm**.

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