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.

Options

A. `40 sqrt 3 cm^2`
B. `49 sqrt 3cm^2`
C. `48 sqrt 6cm^2`
D. `58 sqrt 6cm^2`

Answer

To find the area of the parallelogram with adjacent sides of lengths 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 \( a = 10 \) cm
- Side \( b = 12 \) cm
- Diagonal \( c = 14 \) cm

### Step 2: Calculate the semi-perimeter of triangle ABD
The semi-perimeter \( s \) of triangle ABD can be calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{10 + 12 + 14}{2} = \frac{36}{2} = 18 \text{ cm} \]

### Step 3: Use Heron's formula to find the area of triangle ABD
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 of \( s \), \( a \), \( b \), and \( c \):
\[ A = \sqrt{18(18-10)(18-12)(18-14)} \]
Calculating each term:
\[ A = \sqrt{18 \times 8 \times 6 \times 4} \]

### Step 4: Simplify the expression inside the square root
Calculating the product:
\[ A = \sqrt{18 \times 8 \times 6 \times 4} = \sqrt{3456} \]
To simplify \( \sqrt{3456} \):
\[ \sqrt{3456} = \sqrt{576 \times 6} = 24\sqrt{6} \text{ cm}^2 \]

### Step 5: Calculate the area of the parallelogram
The area of the parallelogram is twice the area of triangle ABD:
\[ \text{Area of parallelogram} = 2 \times A = 2 \times 24\sqrt{6} = 48\sqrt{6} \text{ cm}^2 \]

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

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