Heron's Formula:
[tex]\[ s = \frac{a + b + c}{2} \][/tex]
[tex]\[ A = \sqrt{s(s-a)(s-b)(s-c)} \][/tex]

Use Heron's formula to find the area, in square yards, of [tex]\(\triangle ABC\)[/tex].



Answer :

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.