A parallelogram has sides 30 cm and 20 cm and one of its diagonal is 40 cm long. Then its area is :

Options

A. `75sqrt(5) cm^(2) `
B. `245 cm^(2) `
C. ` 150sqrt(15) cm^(2) `
D. `300 cm^(2) `

Answer

To find the area of the parallelogram given its sides and one diagonal, we can use Heron's formula. Here’s a step-by-step solution:

### Step 1: Identify the given values
We have a parallelogram with:
- Side \( a = 30 \) cm
- Side \( b = 20 \) cm
- Diagonal \( d = 40 \) cm

### Step 2: Split the parallelogram into two triangles
The diagonal divides the parallelogram into two triangles, \( \triangle ABC \) and \( \triangle ADC \). Both triangles share the diagonal \( AC \).

### Step 3: Calculate the semi-perimeter of triangle \( ABC \)
The semi-perimeter \( s \) of triangle \( ABC \) is calculated as follows:
\[ s = \frac{a + b + d}{2} = \frac{30 + 20 + 40}{2} = \frac{90}{2} = 45 \text{ cm} \]

### Step 4: Use Heron's formula to find the area of triangle \( ABC \)
Heron's formula states that the area \( A \) of a triangle can be calculated using:
\[ A = \sqrt{s(s-a)(s-b)(s-d)} \]
Substituting the values we have:
\[ A = \sqrt{45(45-30)(45-20)(45-40)} \]
Calculating each term:
- \( s - a = 45 - 30 = 15 \)
- \( s - b = 45 - 20 = 25 \)
- \( s - d = 45 - 40 = 5 \)

Now substituting these values into the formula:
\[ A = \sqrt{45 \times 15 \times 25 \times 5} \]

### Step 5: Simplify the expression
Calculating the product inside the square root:
\[ 45 = 3^2 \times 5, \quad 15 = 3 \times 5, \quad 25 = 5^2, \quad 5 = 5 \]
Thus,
\[ A = \sqrt{(3^2 \times 5) \times (3 \times 5) \times (5^2) \times (5)} = \sqrt{3^3 \times 5^5} \]

### Step 6: Calculate the area of triangle \( ABC \)
Taking the square root:
\[ A = 3^{3/2} \times 5^{5/2} = 3\sqrt{3} \times 25\sqrt{5} = 75\sqrt{15} \text{ cm}^2 \]

### Step 7: Calculate the area of the parallelogram
Since the area of the parallelogram is twice the area of one triangle:
\[ \text{Area of parallelogram} = 2 \times A = 2 \times 75\sqrt{15} = 150\sqrt{15} \text{ cm}^2 \]

### Final Answer
The area of the parallelogram is \( 150\sqrt{15} \text{ cm}^2 \).

---