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

s = $\frac{a+b+c}{2}$

A = $\sqrt{\left(\frac{a+b+c}{2}\right)\left(\right(\frac{a+b+c}{2}-a\left)\right(\frac{a+b+c}{2}-b\left)\right(\frac{a+b+c}{2}-c)}$

A = $\sqrt{\left(\frac{a+b+c}{2}\right)\left(\frac{a+b+c-2a}{2}\right)\left(\frac{a+b+c-2b}{2}\right)\left(\frac{a+b+c-2c}{2}\right)}$

=$\frac{1}{4}\sqrt{(a+b+c)\left(\right(a+b+c)-2a)\left(\right(a+b+c)-2b)\left(\right(a+b+c)-2c}$

A = $\frac{1}{4}(\sqrt{p(p-2a)(p-2b)(p-2c)}$)
$\therefore$ $\frac{\sqrt{p(p-2a)(p-2b)(p-2c)}}{A}$ = 4