Chapter 13, Problem 1PS

Area Heron's Formula states that the area of a triangle with sides of lengths $a$, $b$, and $c$ is given by

$A=\sqrt{s(s-a)(s-b)(s-c)}$

where $s=\frac{a+b+c}{2}$, as shown in the figure.

(a) Use Heron's Formula to find the area of the triangle with vertices $(0,0)$, $(3,4)$, and $(6,0)$.

(b) Show that among all triangles having a fixed perimeter, the triangle with the largest area is a equilateral triangle.

(c) Show that among all triangles having a fixed area, the triangle with the smallest perimeter is an equilateral triangle.