Question

A triangle has sides $6,7$ and $8$. The line through its incentre parallel to the shortest side is drawn to meet the other two sides at $P$ and $Q$. Then the value of $\left[PQ\right]$ is: (where $[.]$ denotes $G.I.F$.)

Answer 1

Let $a=6,b=7,c=8$
So the smallest side is $BC=a=6$
$2s=a+b+c=6+7+8$
$s=\frac{21}{2}$
We know that $\Delta =r\times s=\frac{1}{2}\times BC\times h$
$\therefore \frac{r\times 21}{2}=\frac{6\times h}{2}$
$\Rightarrow \frac{r}{h}=\frac{2}{7}$
Now $\Delta APQ$ and $\Delta ABC$ are similar
$\Rightarrow \frac{h-r}{h}=\frac{PQ}{BC}$
$\Rightarrow \frac{h-r}{h}=\frac{PQ}{6}$
$\Rightarrow 1-\frac{r}{h}=\frac{PQ}{6}$
$\Rightarrow PQ=\frac{30}{7}\Rightarrow \left[PQ\right]=4$