To find the area of the triangle with sides of lengths 56 cm, 65 cm, and 52 cm, we can use Heron's formula. Here is the step-by-step process:
1. Determine the semi-perimeter (s):
The semi-perimeter of a triangle is half the perimeter of the triangle. It is calculated as:
[tex]\[
s = \frac{a + b + c}{2}
\][/tex]
Given:
[tex]\[
a = 56 \, \text{cm}, \, b = 65 \, \text{cm}, \, c = 52 \, \text{cm}
\][/tex]
Substituting these values in:
[tex]\[
s = \frac{56 + 65 + 52}{2} = \frac{173}{2} = 86.5 \, \text{cm}
\][/tex]
2. Use Heron's formula to find the area (A):
Heron's formula for the area of a triangle is:
[tex]\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\][/tex]
Substituting [tex]\( s = 86.5 \, \text{cm} \)[/tex]:
[tex]\[
A = \sqrt{86.5 (86.5 - 56) (86.5 - 65) (86.5 - 52)}
\][/tex]
Calculate the individual terms:
[tex]\[
s - a = 86.5 - 56 = 30.5
\][/tex]
[tex]\[
s - b = 86.5 - 65 = 21.5
\][/tex]
[tex]\[
s - c = 86.5 - 52 = 34.5
\][/tex]
Now plug these back into the formula:
[tex]\[
A = \sqrt{86.5 \times 30.5 \times 21.5 \times 34.5}
\][/tex]
3. Calculate the product inside the square root:
[tex]\[
86.5 \times 30.5 \times 21.5 \times 34.5 \approx 1,956,920.575
\][/tex]
4. Take the square root:
[tex]\[
A \approx \sqrt{1,956,920.575} \approx 1,398.9 \, \text{cm}^2
\][/tex]
Therefore, the area of the triangle is approximately:
[tex]\[
A \approx 1398.9 \, \text{cm}^2
\][/tex]
