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 will follow these steps:

### Step 1: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
where \( a = 56 \, \text{cm} \), \( b = 60 \, \text{cm} \), and \( c = 52 \, \text{cm} \).

Calculating \( s \):
\[ s = \frac{56 + 60 + 52}{2} = \frac{168}{2} = 84 \, \text{cm} \]

### Step 2: Calculate the area using Heron's formula
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s(s - a)(s - b)(s - c)} \]

Substituting the values:
\[ A = \sqrt{84(84 - 56)(84 - 60)(84 - 52)} \]

Calculating each term:
- \( s - a = 84 - 56 = 28 \)
- \( s - b = 84 - 60 = 24 \)
- \( s - c = 84 - 52 = 32 \)

So, we have:
\[ A = \sqrt{84 \times 28 \times 24 \times 32} \]

### Step 3: Factor the numbers to simplify
To simplify \( \sqrt{84 \times 28 \times 24 \times 32} \), we can factor each number:
- \( 84 = 2^2 \times 3 \times 7 \)
- \( 28 = 2^2 \times 7 \)
- \( 24 = 2^3 \times 3 \)
- \( 32 = 2^5 \)

Now, multiplying these factors together:
\[ 84 \times 28 \times 24 \times 32 = (2^2 \times 3 \times 7) \times (2^2 \times 7) \times (2^3 \times 3) \times (2^5) \]

Combining the powers of 2:
\[ = 2^{2+2+3+5} \times 3^{1+1} \times 7^{1+1} = 2^{12} \times 3^2 \times 7^2 \]

### Step 4: Calculate the square root
Now we can take the square root:
\[ A = \sqrt{2^{12} \times 3^2 \times 7^2} = 2^{6} \times 3 \times 7 = 64 \times 3 \times 7 \]

Calculating:
\[ 64 \times 3 = 192 \]
\[ 192 \times 7 = 1344 \]

Thus, the area \( A \) is:
\[ A = 1344 \, \text{cm}^2 \]

### Final Answer
The area of the triangle is \( 1344 \, \text{cm}^2 \).

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