Question

right triangle relationships and trigonometry
what is the area of triangle abc? round to the nearest square unit.
herons formula: area = \\(\sqrt{s(s - a)(s - b)(s - c)})

Explanation:

Step1: Calculate semi - perimeter s

First, find the semi - perimeter $s$ of the triangle with side lengths $a = 8$, $b = 10$, $c = 16$. The formula for the semi - perimeter is $s=\frac{a + b + c}{2}$. So, $s=\frac{8+10 + 16}{2}=\frac{34}{2}=17$.

Step2: Apply Heron's formula

Heron's formula is $A=\sqrt{s(s - a)(s - b)(s - c)}$. Substitute $s = 17$, $a = 8$, $b = 10$, $c = 16$ into the formula:
\[

$$\begin{align*} A&=\sqrt{17(17 - 8)(17 - 10)(17 - 16)}\\ &=\sqrt{17\times9\times7\times1}\\ &=\sqrt{1071}\\ &\approx32.726 \end{align*}$$

\]
Rounding to the nearest square unit, we get 33 square units.

Answer:

33 square units