The sides of a triangle are `56 cm`, `60 cm` and `52 cm` long. Then, the area of the triangle is

Options

A. 1322 `cm^(2)`
B. 1311 `cm^(2)`
C. 1344 `cm^(2)`
D. 1392 `cm^(2)`

Explanation

To find the area of a triangle with sides measuring 56 cm, 60 cm, and 52 cm using Heron's formula, we follow these steps:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( A = 56 \, \text{cm} \)
- \( B = 60 \, \text{cm} \)
- \( C = 52 \, \text{cm} \)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{A + B + C}{2} \]
Substituting the values:
\[ s = \frac{56 + 60 + 52}{2} = \frac{168}{2} = 84 \, \text{cm} \]

### Step 3: Use Heron's formula to find the area (A)
Heron's formula states that the area \( A \) of the triangle can be calculated as:
\[ A = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting the values:
\[ A = \sqrt{84(84 - 56)(84 - 60)(84 - 52)} \]
Calculating each term:
- \( s - A = 84 - 56 = 28 \)
- \( s - B = 84 - 60 = 24 \)
- \( s - C = 84 - 52 = 32 \)

Now substituting these values back into the area formula:
\[ A = \sqrt{84 \times 28 \times 24 \times 32} \]

### Step 4: Calculate the product inside the square root
First, calculate the product:
- \( 84 \times 28 = 2352 \)
- \( 24 \times 32 = 768 \)

Now, multiply these two results:
\[ 2352 \times 768 \]

Calculating \( 2352 \times 768 \):
\[ 2352 \times 768 = 1,804,736 \]

### Step 5: Find the square root
Now, we find the square root:
\[ A = \sqrt{1,804,736} \approx 1344 \, \text{cm}^2 \]

### Conclusion
Thus, the area of the triangle is approximately:
\[ \text{Area} \approx 1344 \, \text{cm}^2 \]