Question

If length of the side $BC$ of a $\Delta ABC$ is $4$ cm and $\angle BAC={120}^{\circ },$ then the distance between the incenter and excentre of the circle touching the side $BC$ is:

• A. $8$
• B. $9$
• C. $7$
• D. $6$

Answer 1

The correct option is A $8$

Given: $BC$ is $4$ cm and $\angle BAC={120}^{\circ }$
We know, if $I,I_1$ are the incentre and excentre respectively with respect to angle $A$, then:
$I{I}_1=A{I}_1-AI=(r_1-r)\text{cosec}\frac{A}{2}$
$=4R\sin \frac{A}{2}(\cos \frac{B}{2}\cos \frac{C}{2}-\sin \frac{B}{2}\sin \frac{C}{2})\text{cosec}\frac{A}{2}$
$=4R\cos \left(\frac{B+C}{2}\right)=4R\sin \frac{A}{2}=2R\frac{\sin A}{\cos A/2}$
$=\frac{a}{\cos A/2}=\frac{4}{\cos {60}^{\circ }}=8$