A triangle has sides of length 10, 12, and 14. 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:

• $$a = 10$$
• $$b = 12$$
• $$c = 14$$

First, calculate the semi-perimeter:

\[ s = \frac{10+12+14}{2} = \frac{36}{2} = 18 \]

Now, apply Heron's formula:

\[ \text{Area} = \sqrt{18(18-10)(18-12)(18-14)} \]

\[ \text{Area} = \sqrt{18(8)(6)(4)} \]

\[ \text{Area} = \sqrt{18 \times 192} \]

\[ \text{Area} = \sqrt{3456} \]

\[ \text{Area} = \sqrt{576 \times 6} = 24\sqrt{6} \]

Ответ: $$24\sqrt{6}$$