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 an equilateral triangle.

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

Figure for 1