Answer:
Perimeter: p = 30 units
Semiperimeter: s = 15 units
Area: A ≈ 39.7 square units
Step-by-step explanation:
Perimeter
The perimeter of a two-dimensional shape is the total distance around its outer edge.
Therefore, the perimeter (p) of the given triangle is the sum of its side lengths:
[tex]p=8+12+10 \\\\ p=30\; \sf units[/tex]
So, the perimeter of the triangle is:
[tex]\Large\boxed{\boxed{p=30\; \sf units}}[/tex]
[tex]\dotfill[/tex]
Semiperimeter
The semiperimeter is half the perimeter of a polygon.
Therefore, the semiperimeter (s) of the given triangle can be calculated by dividing its perimeter by 2:
[tex]s=\dfrac{p}{2} \\\\\\s=\dfrac{30}{2} \\\\\\ s=15\; \sf units[/tex]
So, the semiperimeter of the triangle is:
[tex]\Large\boxed{\boxed{s=15\; \sf units}}[/tex]
[tex]\dotfill[/tex]
Area
Heron's Formula allows us to find the area of a triangle in terms of its side lengths.
[tex]\boxed{\begin{array}{l}\underline{\textsf{Heron's Formula}}\\\\A=\sqrt{s(s-a)(s-b)(s-c)}\\\\\textsf{where:}\\ \phantom{ww}\bullet \;\textsf{$A$ is the area of the triangle.} \\ \phantom{ww}\bullet \;\textsf{$a, b$ and $c$ are the side lengths of the triangle.} \\ \phantom{ww}\bullet \;\textsf{$s$ is half the perimeter.}\end{array}}[/tex]
In this case:
Substitute the values into the formula and solve for A:
[tex]A=\sqrt{15(15-8)(15-12)(15-10)} \\\\ A=\sqrt{15(7)(3)(5)} \\\\ A=\sqrt{1575} \\\\ A=39.68626966596... \\\\A=39.7\; \sf square\;units\;(nearest\;tenth)[/tex]
Therefore, the area of the triangle is:
[tex]\Large\boxed{\boxed{A \approx 39.7\; \sf square\;units}}[/tex]
