Question

herons formula
area = \sqrt{s(s - a)(s - b)(s - c)}
use the triangle pictured to calculate the following measurements. then, calculate the area rounded to the nearest tenth using herons formula
perimeter p = units
semi - perimeter s = units
area a = square units

Explanation:

Step1: Calculate perimeter

The perimeter $p$ of a triangle with side - lengths $a = 8$, $b = 10$, and $c = 12$ is the sum of the side - lengths. So, $p=a + b + c=8 + 10+12=30$ units.

Step2: Calculate semi - perimeter

The semi - perimeter $s$ is half of the perimeter. So, $s=\frac{p}{2}=\frac{30}{2}=15$ units.

Step3: Calculate area using Heron's formula

Heron's formula is $A=\sqrt{s(s - a)(s - b)(s - c)}$. Substitute $s = 15$, $a = 8$, $b = 10$, and $c = 12$ into the formula:
\[

$$\begin{align*} A&=\sqrt{15(15 - 8)(15 - 10)(15 - 12)}\\ &=\sqrt{15\times7\times5\times3}\\ &=\sqrt{1575}\\ &\approx39.7 \end{align*}$$

\]

Answer:

Perimeter $p = 30$ units
Semi - perimeter $s = 15$ units
Area $A\approx39.7$ square units