Two adjacent sides of a parallelogram are 10 cm and 12 cm. If its one diagonal is 14 cm long, find the area of the parallelogram.

Answer

To find the area of the parallelogram with two adjacent sides measuring 10 cm and 12 cm, and one diagonal measuring 14 cm, we can follow these steps:

### Step 1: Identify the sides and diagonal
Let the sides of the parallelogram be:
- Side A = 12 cm
- Side B = 10 cm
- Diagonal C = 14 cm

### Step 2: Calculate the semi-perimeter (s) of triangle ABC
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{A + B + C}{2} \]
Substituting the values:
\[ s = \frac{12 + 10 + 14}{2} = \frac{36}{2} = 18 \text{ cm} \]

### Step 3: Apply Heron's formula to find the area of triangle ABC
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting the values:
\[ A = \sqrt{18(18 - 12)(18 - 10)(18 - 14)} \]
Calculating each term:
- \( s - A = 18 - 12 = 6 \)
- \( s - B = 18 - 10 = 8 \)
- \( s - C = 18 - 14 = 4 \)

Now substituting these values into the formula:
\[ A = \sqrt{18 \times 6 \times 8 \times 4} \]

### Step 4: Simplify the expression
Calculating the product:
\[ 18 \times 6 = 108 \]
\[ 108 \times 8 = 864 \]
\[ 864 \times 4 = 3456 \]
Thus, we have:
\[ A = \sqrt{3456} \]

### Step 5: Factor and simplify the square root
To simplify \( \sqrt{3456} \):
\[ 3456 = 576 \times 6 \]
Since \( \sqrt{576} = 24 \), we get:
\[ A = 24\sqrt{6} \text{ cm}^2 \]

### Step 6: Calculate the area of the parallelogram
The area of the parallelogram is twice the area of triangle ABC:
\[ \text{Area of parallelogram} = 2 \times 24\sqrt{6} = 48\sqrt{6} \text{ cm}^2 \]

### Final Answer
The area of the parallelogram is \( 48\sqrt{6} \text{ cm}^2 \).

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