Sure, let's go through the steps to find the area of triangle [tex]\( \triangle ABC \)[/tex] using Heron's formula. The lengths of the sides of the triangle are:
[tex]\( a = 5 \)[/tex] yards, [tex]\( b = 6 \)[/tex] yards, and [tex]\( c = 7 \)[/tex] yards.
1. Calculate the semi-perimeter [tex]\( s \)[/tex]:
The semi-perimeter [tex]\( s \)[/tex] is given by:
[tex]\[
s = \frac{a + b + c}{2}
\][/tex]
Plug in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
s = \frac{5 + 6 + 7}{2} = \frac{18}{2} = 9 \text{ yards}
\][/tex]
2. Calculate the area [tex]\( A \)[/tex] using Heron's formula:
Heron's formula for the area [tex]\( A \)[/tex] is:
[tex]\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\][/tex]
Substitute the values from the previous calculations:
[tex]\[
A = \sqrt{9(9 - 5)(9 - 6)(9 - 7)}
\][/tex]
Simplify the expression inside the square root:
[tex]\[
A = \sqrt{9 \times 4 \times 3 \times 2}
\][/tex]
Continue simplifying:
[tex]\[
A = \sqrt{9 \times 24} = \sqrt{216}
\][/tex]
Take the square root of 216:
[tex]\[
A \approx 14.696938456699069 \text{ square yards}
\][/tex]
Therefore, the area of [tex]\(\triangle ABC\)[/tex] is approximately 14.696938456699069 square yards.
