A parallelogram has sides 60 m and 40 m and one of its diagonals is 80 m long. Its area is.
एक समांतर चतुर्भुज की भुजाएँ 60m और 40m हैं और उसका एक विकर्ण 80m लंबा है। उसका क्षेत्रफल कितना होगा?

Options

A. `450 sqrt(15) m^(2)`
B. `500 sqrt(15)m^(2)`
C. `600 sqrt(15) m^(2)`
D. `400 sqrt(15) m^(2)`

Explanation

To find the area of the parallelogram with sides 60 m and 40 m and a diagonal of 80 m, we will use Heron's formula. Here’s a step-by-step solution:

### Step 1: Identify the sides and diagonal
We have:
- Side AB = 60 m
- Side AD = 40 m
- Diagonal AC = 80 m

### Step 2: Calculate the semi-perimeter
The semi-perimeter (s) of triangle ABC can be calculated using the formula:
\[ s = \frac{AB + BC + AC}{2} \]
Since BC is equal to AD (as opposite sides of a parallelogram are equal), we have:
\[ s = \frac{60 + 40 + 80}{2} = \frac{180}{2} = 90 \, m \]

### Step 3: Apply Heron's formula
Heron's formula for the area (A) of a triangle is given by:
\[ A = \sqrt{s(s - a)(s - b)(s - c)} \]
Where:
- \( a = AB = 60 \, m \)
- \( b = AC = 80 \, m \)
- \( c = AD = 40 \, m \)

Substituting the values into the formula:
\[ A = \sqrt{90(90 - 60)(90 - 80)(90 - 40)} \]
\[ A = \sqrt{90 \times 30 \times 10 \times 50} \]

### Step 4: Simplify the expression
Calculating the values inside the square root:
\[ A = \sqrt{90 \times 30 \times 10 \times 50} \]
\[ = \sqrt{1350000} \]

### Step 5: Factor the expression
We can simplify \( 1350000 \):
\[ 1350000 = 135 \times 10000 = 135 \times 100^2 \]
\[ = 135 \times 100^2 = 10000 \times 135 \]
So,
\[ A = 100 \sqrt{135} \]

### Step 6: Calculate the area of the parallelogram
Since the diagonal divides the parallelogram into two equal triangles, the total area of the parallelogram will be:
\[ \text{Area of Parallelogram} = 2 \times A = 2 \times 100 \sqrt{135} = 200 \sqrt{135} \]

### Step 7: Final simplification
We can simplify \( \sqrt{135} \):
\[ \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15} \]
Thus, the area becomes:
\[ \text{Area of Parallelogram} = 200 \times 3\sqrt{15} = 600\sqrt{15} \]

### Final Answer
The area of the parallelogram is \( 600\sqrt{15} \, m^2 \).

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