What is the area of the triangle whose side are 84 m , 80 m and 52 m ?

Options

A. 1620 sq. m
B. 2016 sq. m
C. 1818 sq. m
D. none of these

Answer

To find the area of a triangle with sides measuring 84 m, 80 m, and 52 m, we can use Heron's formula. Here’s a step-by-step solution:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( A = 84 \) m
- \( B = 80 \) m
- \( C = 52 \) m

### 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:
\[ S = \frac{84 + 80 + 52}{2} = \frac{216}{2} = 108 \text{ m} \]

### Step 3: Apply Heron's formula
Heron's formula for the area \( A \) of the triangle is given by:
\[ \text{Area} = \sqrt{S \times (S - A) \times (S - B) \times (S - C)} \]
Substituting the values we calculated:
\[ \text{Area} = \sqrt{108 \times (108 - 84) \times (108 - 80) \times (108 - 52)} \]
Calculating each term:
- \( S - A = 108 - 84 = 24 \)
- \( S - B = 108 - 80 = 28 \)
- \( S - C = 108 - 52 = 56 \)

### Step 4: Substitute the values into the formula
Now substituting these values back into the formula:
\[ \text{Area} = \sqrt{108 \times 24 \times 28 \times 56} \]

### Step 5: Calculate the product under the square root
Calculating the product:
- First, calculate \( 24 \times 28 = 672 \)
- Next, calculate \( 672 \times 56 = 37632 \)
- Finally, calculate \( 108 \times 37632 = 4064256 \)

### Step 6: Take the square root
Now, take the square root of the product:
\[ \text{Area} = \sqrt{4064256} = 2016 \text{ m}^2 \]

### Conclusion
Thus, the area of the triangle is:
\[ \text{Area} = 2016 \text{ m}^2 \]

---