A triangle has sides of length 8, 10, and 12. 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:

• $$a = 8$$
• $$b = 10$$
• $$c = 12$$

First, calculate the semi-perimeter:

\[ s = \frac{8+10+12}{2} = \frac{30}{2} = 15 \]

Now, apply Heron's formula:

\[ \text{Area} = \sqrt{15(15-8)(15-10)(15-12)} \]

\[ \text{Area} = \sqrt{15(7)(5)(3)} \]

\[ \text{Area} = \sqrt{15 \times 105} \]

\[ \text{Area} = \sqrt{1575} \]

\[ \text{Area} = \sqrt{225 \times 7} = 15\sqrt{7} \]

Ответ: $$15\sqrt{7}$$