A triangle has sides of length 12, 14, and 16. Use Heron's formula to find the area.

Heron's formula for the area of a triangle with sides $$a, b, c$$ is: Area $$= \sqrt{s(s-a)(s-b)(s-c)}$$, where $$s$$ is the semi-perimeter, $$s = \frac{a+b+c}{2}$$.

Given:

• $$a = 12$$
• $$b = 14$$
• $$c = 16$$

First, calculate the semi-perimeter:

\[ s = \frac{12+14+16}{2} = \frac{42}{2} = 21 \]

Now, apply Heron's formula:

\[ \text{Area} = \sqrt{21(21-12)(21-14)(21-16)} \]

\[ \text{Area} = \sqrt{21(9)(7)(5)} \]

\[ \text{Area} = \sqrt{21 \times 315} \]

\[ \text{Area} = \sqrt{6615} \]

\[ \text{Area} = \sqrt{9 \times 735} = 3\sqrt{735} \]

\[ \text{Area} = 3\sqrt{49 \times 15} = 3 \times 7 \sqrt{15} = 21\sqrt{15} \]

Ответ: $$21\sqrt{15}$$