Question

The angular points of a triangle are $A(–1,–7),B(5,1)$ and $C(1,4)$. The equation of the bisector of the angle $\angle ABC$ is

• A. $x=7y+2$
• B. $7y=x+2$
• C. $y=7x+2$
• D. $7x=y+2$

Answer 1

The correct option is B $7y=x+2$


$AB=\sqrt{36+64}=10BC=\sqrt{16+9}=5CA=\sqrt{121+4}=\sqrt{125}$
Angle bisector divides the opposite side of a triangle into segements that are proportional to the other two sides
$\therefore \frac{AD}{DC}=\frac{10}{5}=2$
So the coordinate of the D is
$D(\frac{-1+2}{3},\frac{-7+8}{3})=D(\frac{1}{3},\frac{1}{3})$

Equation of BD is $y-1=\frac{1-\frac{1}{3}}{5-\frac{1}{3}}(x-5)$
$\Rightarrow y-1=\frac{2}{14}(x-5)\Rightarrow 7y-7=x-5\Rightarrow x+2=7y$