Question

Suppose p denotes the perimeter of a triangle with sides a, b and c and s is the semi perimeter. If A denotes the area of the triangle, $\frac{\sqrt{p(p-2a)(p-2b)(p-2c)}}{A}$ = __.

Answer 1

We know that Area = $\sqrt{s(s-a)(s-b)(s-c)}$
A = $\sqrt{\left(\frac{a+b+c}{2}\right)\left(\frac{a+b-c}{2}\right)\left(\frac{a-b+c}{2}\right)\left(\frac{-a+b+c}{2}\right)}$
A = $\frac{1}{4}(\sqrt{(a+b+c)(a+b-c)(a-b+c)(-a+b+c)}$)
A = $\frac{1}{4}(\sqrt{(a+b+c)\left(\right(a+b+c)-2c)\left(\right(a+b+c)-2b)\left(\right(a+b+c)-2a)}$)
A = $\frac{1}{4}(\sqrt{p(p-2a)(p-2b)(p-2c)}$)
$\therefore$ $\frac{\sqrt{p(p-2a)(p-2b)(p-2c)}}{A}$ = 4