Question

If the altitudes of a triangle be $3,4,6$, then find its in-radius.

Answer 1

Let the altitudes be $h_1=3,h_2=4,h_3=6$

$\Delta =\frac{1}{2}{h}_{1}a⟹\frac{a}{\Delta }=\frac{2}{h_1}.....\left(1\right)$

Similarly,$\frac{b}{\Delta }=\frac{2}{h_2}.....\left(2\right),\frac{c}{\Delta }=\frac{2}{h_3}......\left(3\right)$

$\left(1\right)+\left(2\right)+\left(3\right)⟹\frac{a+b+c}{\Delta }=2(\frac{1}{h_1}+\frac{1}{h_2}+\frac{1}{h_3})$

$\frac{2s}{\Delta }=2(\frac{1}{3}+\frac{1}{4}+\frac{1}{6})$

$\frac{1}{r}=\frac{3}{4}$

$r=\frac{4}{3}$