The sides of parallelogram are 12 m and 17 m respectively . If one of the diagonals is 25 m long the area of parallelogram is :
A)`150 m^(2)`
B)`180 m^(2)`
C)`160 m^(2)`
D)`190 m^(2)`

Options

A. `150 m^(2)`
B. `180 m^(2)`
C. `160 m^(2)`
D. `190 m^(2)`

Answer

To find the area of the parallelogram with sides 12 m and 17 m, and one diagonal measuring 25 m, we can use Heron's formula to first calculate the area of one of the triangles formed by the diagonal.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
The triangle formed by the diagonal consists of the sides of the parallelogram and the diagonal. The sides are:
- Side A = 12 m
- Side B = 17 m
- Diagonal (Side C) = 25 m

2. **Calculate the semi-perimeter (s)**:
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{A + B + C}{2} \]
Substituting the values:
\[ s = \frac{12 + 17 + 25}{2} = \frac{54}{2} = 27 \text{ m} \]

3. **Apply 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:
\[ A = \sqrt{27 \times (27 - 12) \times (27 - 17) \times (27 - 25)} \]
Simplifying:
\[ A = \sqrt{27 \times 15 \times 10 \times 2} \]

4. **Calculate the area**:
First, calculate the product inside the square root:
\[ 27 \times 15 = 405 \]
\[ 405 \times 10 = 4050 \]
\[ 4050 \times 2 = 8100 \]
Now, take the square root:
\[ A = \sqrt{8100} = 90 \text{ m}^2 \]

5. **Calculate the area of the parallelogram**:
Since the diagonal divides the parallelogram into two equal triangles, the area of the parallelogram is:
\[ \text{Area of parallelogram} = 2 \times \text{Area of triangle ABD} = 2 \times 90 = 180 \text{ m}^2 \]

### Final Answer:
The area of the parallelogram is **180 m²**.