What is the exact area of a triangle with sides measuring 7, 8, and 9 in simplest form?

To find the area of a triangle with side lengths 7, 8, and 9, we can apply Heron's formula, which is useful for calculating the area when the lengths of all three sides are known.

First, we calculate the semi-perimeter (s) of the triangle:

[

s = \frac{a + b + c}{2} = \frac{7 + 8 + 9}{2} = 12

]

Next, we use Heron's formula, which states that the area (A) of the triangle can be calculated as follows:

[

A = \sqrt{s(s-a)(s-b)(s-c)}

]

Substituting the values:

[

A = \sqrt{12(12-7)(12-8)(12-9)}

]

[

A = \sqrt{12 \times 5 \times 4 \times 3}

]

[

A = \sqrt{12 \times 60}

]

[

A = \sqrt{720}

]

To simplify (\sqrt{720}), we can factor it as follows:

[

720 = 144 \times 5

]

Thus, we can express the