Q.
Consider a triangle $ABC$ with vertex $A (2,-4)$. The internal bisectors of the angles $B$ and $C$ are $x+y=2$ and $x-3 y=6$ respectively. Let the two bisectors meet at $I$.
Measure of the angle BIC equals
Straight Lines
Solution:
$\text { From figure, } 135^{\circ}=\theta+\theta_1 \text { where } \tan \theta_1=\frac{1}{3} $
$\theta=135^{\circ}-\theta_1 $
$\tan \theta=\tan \left(135^{\circ}-\theta_1\right)$
$=-\tan \left(45^{\circ}+\theta_1\right)=-\left(\frac{1+\tan \theta_1}{1-\tan \theta_1}\right)--\left(\frac{4 / 3}{2 / 3}\right)=-2 $
$\Rightarrow \theta=\pi-\tan ^{-1} 2$
