The adjacent sides of a parallelogram are 21 cm and 28 cm. If its one diagonal is 35 cm, find the area of the parallelogram.

Answer

To find the area of the parallelogram with adjacent sides of 21 cm and 28 cm, and a diagonal of 35 cm, we can use the following steps:

### Step 1: Identify the sides and diagonal
Let the sides of the parallelogram be \( AB = 21 \, \text{cm} \) and \( AD = 28 \, \text{cm} \). The diagonal \( AC = 35 \, \text{cm} \).

### Step 2: Calculate the semi-perimeter of triangle ABC
The semi-perimeter \( s \) is given by the formula:
\[ s = \frac{AB + AC + BC}{2} \]
Here, \( BC \) is equal to \( AD \), so we have:
\[ s = \frac{21 + 35 + 28}{2} = \frac{84}{2} = 42 \, \text{cm} \]

### Step 3: Use Heron's formula to find the area of triangle ABC
Heron's formula for the area \( \Delta \) of a triangle is:
\[ \Delta = \sqrt{s(s - a)(s - b)(s - c)} \]
where \( a = AB = 21 \, \text{cm} \), \( b = AC = 35 \, \text{cm} \), and \( c = AD = 28 \, \text{cm} \).

Now, we calculate:
\[ \Delta = \sqrt{42(42 - 21)(42 - 35)(42 - 28)} \]
Calculating each term:
\[ 42 - 21 = 21, \quad 42 - 35 = 7, \quad 42 - 28 = 14 \]
So, we have:
\[ \Delta = \sqrt{42 \times 21 \times 7 \times 14} \]

### Step 4: Simplify the expression
Now, we can simplify:
\[ 42 = 7 \times 6, \quad 21 = 7 \times 3, \quad 14 = 7 \times 2 \]
Thus,
\[ \Delta = \sqrt{(7 \times 6) \times (7 \times 3) \times (7) \times (7 \times 2)} = \sqrt{7^4 \times 6 \times 3 \times 2} \]
Calculating \( 6 \times 3 \times 2 = 36 \):
\[ \Delta = \sqrt{7^4 \times 36} = 7^2 \times \sqrt{36} = 49 \times 6 = 294 \, \text{cm}^2 \]

### Step 5: Calculate the area of the parallelogram
The area of the parallelogram is twice the area of triangle ABC:
\[ \text{Area of parallelogram} = 2 \times \Delta = 2 \times 294 = 588 \, \text{cm}^2 \]

### Final Answer
The area of the parallelogram is \( 588 \, \text{cm}^2 \).

---