A parallelogram has two sides 60 m and 25 m and a diagonal 65 m long. The area of the parallelogram is :

Options

A. 1000 `m^(2)`
B. 1400 `m^(2)`
C. 1600 `m^(2)`
D. 1500 `m^(2)`

Answer

To find the area of the parallelogram with sides of lengths 60 m and 25 m, and a diagonal of length 65 m, we can use Heron's formula to first find the area of triangle ACD formed by the diagonal. The area of the parallelogram will then be double that area.

### Step-by-Step Solution:

1. **Identify the sides of triangle ACD**:
- The sides are given as:
- AC = 60 m
- AD = 25 m
- CD (the diagonal) = 65 m

2. **Calculate the semi-perimeter (s)**:
\[ s = \frac{AC + AD + CD}{2} = \frac{60 + 25 + 65}{2} = \frac{150}{2} = 75 \text{ m} \]

3. **Apply Heron's formula**:
- Heron's formula states that the area (A) of a triangle can be calculated using:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
- Here, \( a = 60 \, m \), \( b = 25 \, m \), \( c = 65 \, m \).

4. **Substitute the values into Heron's formula**:
\[ A = \sqrt{75(75 - 60)(75 - 25)(75 - 65)} \]
\[ A = \sqrt{75(15)(50)(10)} \]

5. **Calculate the individual terms**:
- First, calculate \( 75 \times 15 = 1125 \)
- Next, calculate \( 1125 \times 50 = 56250 \)
- Finally, calculate \( 56250 \times 10 = 562500 \)

6. **Find the square root**:
\[ A = \sqrt{562500} = 750 \text{ m}^2 \]

7. **Calculate the area of the parallelogram**:
- The area of the parallelogram is double the area of triangle ACD:
\[ \text{Area of parallelogram} = 2 \times A = 2 \times 750 = 1500 \text{ m}^2 \]

### Final Answer:
The area of the parallelogram is **1500 m²**.