PQRS is a parallegoram whose adjecent sides PQ = 20 cm and QR = 21 cm one of its diagonals PR = 13 cm , find :
the distance between the shorter sides.

Answer

To find the distance between the shorter sides of the parallelogram PQRS, we can follow these steps:

### Step 1: Understand the Problem
We have a parallelogram PQRS with the following dimensions:
- PQ = 20 cm (shorter side)
- QR = 21 cm (longer side)
- Diagonal PR = 13 cm

### Step 2: Draw a Diagram
Draw the parallelogram PQRS with the given dimensions. Label the sides and the diagonal.

### Step 3: Calculate the Semi-Perimeter
Using the formula for the semi-perimeter (s):
\[ s = \frac{PQ + QR + PR}{2} \]
Substituting the values:
\[ s = \frac{20 + 21 + 13}{2} = \frac{54}{2} = 27 \text{ cm} \]

### Step 4: Apply Heron's Formula to Find the Area of Triangle PQR
Using Heron's formula:
\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
Where \(a = PR\), \(b = PQ\), and \(c = QR\):
\[ \text{Area} = \sqrt{27(27-13)(27-20)(27-21)} \]
Calculating the individual terms:
- \(s - a = 27 - 13 = 14\)
- \(s - b = 27 - 20 = 7\)
- \(s - c = 27 - 21 = 6\)

Now substituting back into the formula:
\[ \text{Area} = \sqrt{27 \times 14 \times 7 \times 6} \]
Calculating the product:
\[ = \sqrt{15876} = 126 \text{ cm}^2 \]

### Step 5: Calculate the Area of Parallelogram PQRS
The area of parallelogram PQRS is twice the area of triangle PQR:
\[ \text{Area of PQRS} = 2 \times 126 = 252 \text{ cm}^2 \]

### Step 6: Use the Area to Find the Height
The area of a parallelogram can also be expressed as:
\[ \text{Area} = \text{Base} \times \text{Height} \]
Here, the base is PQ = 20 cm, and we need to find the height (h):
\[ 252 = 20 \times h \]
Solving for h:
\[ h = \frac{252}{20} = 12.6 \text{ cm} \]

### Conclusion
The distance between the shorter sides (height) of the parallelogram PQRS is **12.6 cm**.

---