To find the area of a triangle given the lengths of its sides, we can use Heron's formula. First, we need to calculate the semi-perimeter (s) of the triangle.
The formula for the semi-perimeter is:
\[ s = \frac{A + B + C}{2} \]
In this case:
- \( A = 125 , \text{ft} \)
- \( B = 160 , \text{ft} \)
- \( C = 135 , \text{ft} \)
Calculating the semi-perimeter:
\[ s = \frac{125 + 160 + 135}{2} = \frac{420}{2} = 210 , \text{ft} \]
Now we can use Heron's formula to find the area (A):
\[ A = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting in the values we have:
\[ A = \sqrt{210 \times (210 - 125) \times (210 - 160) \times (210 - 135)} \]
Calculating each term inside the square root:
- \( s - A = 210 - 125 = 85 \)
- \( s - B = 210 - 160 = 50 \)
- \( s - C = 210 - 135 = 75 \)
Now, plugging these values back in:
\[ A = \sqrt{210 \times 85 \times 50 \times 75} \]
Calculating the product:
First, calculate \( 85 \times 50 \):
\[ 85 \times 50 = 4250 \]
Now calculate \( 4250 \times 75 \):
\[ 4250 \times 75 = 318750 \]
Now we have:
\[ A = \sqrt{210 \times 318750} \]
Calculating the product:
\[ 210 \times 318750 = 66862500 \]
Now take the square root:
\[ A = \sqrt{66862500} \approx 8183.74 , \text{sq ft} \]
Rounding to the nearest hundred square feet gives us:
8100 square feet
Thus, the correct answer from the options is:
8100