A triangle has sides of length 10, 13, and 15. 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:

• $$a = 10$$
• $$b = 13$$
• $$c = 15$$

First, calculate the semi-perimeter:

\[ s = \frac{10+13+15}{2} = \frac{38}{2} = 19 \]

Now, apply Heron's formula:

\[ \text{Area} = \sqrt{19(19-10)(19-13)(19-15)} \]

\[ \text{Area} = \sqrt{19(9)(6)(4)} \]

\[ \text{Area} = \sqrt{19 \times 216} \]

\[ \text{Area} = \sqrt{4104} \]

\[ \text{Area} = \sqrt{36 \times 114} = 6\sqrt{114} \]

Ответ: $$6\sqrt{114}$$