PQRS is a parallegoram whose adjecent sides PQ = 20 cm and QR = 21 cm one of its diagonals QS = 13 cm , find :
the area of the parallelogram PQRS.

Answer

To find the area of the parallelogram PQRS, we can use the properties of triangles and Heron's formula. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Length of side PQ = 20 cm
- Length of side QR = 21 cm
- Length of diagonal QS = 13 cm

### Step 2: Calculate the perimeter of triangle PQS
The perimeter (P) of triangle PQS can be calculated by adding the lengths of its sides:
\[ P = PQ + QS + QR = 20 + 13 + 21 = 54 \, \text{cm} \]

### Step 3: Calculate the semi-perimeter (s)
The semi-perimeter (s) is half of the perimeter:
\[ s = \frac{P}{2} = \frac{54}{2} = 27 \, \text{cm} \]

### Step 4: Use Heron's formula to find the area of triangle PQS
Heron's formula states that the area (A) of a triangle can be calculated using:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( a, b, c \) are the lengths of the sides of the triangle.

In triangle PQS:
- \( a = PQ = 20 \, \text{cm} \)
- \( b = QR = 21 \, \text{cm} \)
- \( c = QS = 13 \, \text{cm} \)

Substituting these values into Heron's formula:
\[ A = \sqrt{27(27-20)(27-21)(27-13)} \]
\[ A = \sqrt{27 \times 7 \times 6 \times 14} \]

### Step 5: Calculate the area of triangle PQS
Now we simplify:
1. Calculate \( 27 \times 7 = 189 \)
2. Calculate \( 6 \times 14 = 84 \)
3. Now multiply: \( 189 \times 84 \)

To simplify calculations:
- \( 189 = 9 \times 21 \)
- \( 84 = 12 \times 7 \)

Thus, \( 189 \times 84 = (9 \times 21) \times (12 \times 7) \).

Now, calculate:
\[ 189 \times 84 = 15876 \]

Now, take the square root:
\[ A = \sqrt{15876} = 126 \, \text{cm}^2 \]

### Step 6: Calculate the area of the parallelogram PQRS
Since the area of the parallelogram is twice the area of triangle PQS:
\[ \text{Area of PQRS} = 2 \times A = 2 \times 126 = 252 \, \text{cm}^2 \]

### Final Answer
The area of the parallelogram PQRS is \( 252 \, \text{cm}^2 \).

---