Question

Three vertices of a triangle ABC are A (-1, 11), B (-9, -8) and C (15, -2). The equation of the angle bisector of angle A is:
(a)$4x-y=7$
(b)$4x+y=7$
(c)$x+4y=7$
(d) $x-4y=7$

Answer 1

First of all, draw a triangle ABC with vertices A (-1, 11), B (-9, -8) and C (15, -2). We are asked to find the equation of angle bisector of angle A. The angle bisector of angle A will pass through angle A and the midpoint of vertices B and C so we have two points A and the midpoint of B and C. We can write the equation of a straight line if we have two points. The equation of a straight line if we have two points say $\left( {{x}_{1}},{{y}_{1}} \right)\And \left( {{x}_{2}},{{y}_{2}} \right)$ is equal to $y-{{y}_{1}}=\left( \dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \right)\left( x-{{x}_{1}} \right)$. Using this equation format, we can write the equation for the angle bisector of angle A.

Complete step-by-step answer:
We have given a triangle ABC with three vertices A (-1, 11), B (-9, -8) and C (15, -2). In the below figure, we have drawn this triangle ABC.

Now, we are going to find the midpoint of BC. We know that, the formula for midpoint of two vertices say $\left( {{x}_{3}},{{y}_{3}} \right)\And \left( {{x}_{4}},{{y}_{4}} \right)$ is equal to:
$\left( \dfrac{{{x}_{3}}+{{x}_{4}}}{2},\dfrac{{{y}_{3}}+{{y}_{4}}}{2} \right)$
Now, taking the points B (-9, -8) and C (15, -2) and using the above midpoint formula we get,
$\begin{aligned} & \left( \dfrac{-9+15}{2},\dfrac{-8-2}{2} \right) \\\ & =\left( \dfrac{6}{2},\dfrac{-10}{2} \right) \\\ & =\left( 3,-5 \right) \\\ \end{aligned}$
Let us name this midpoint as D (3, -5). Now, marking this point on the side BC we get,

Now, we are going to draw the angle bisector of angle A which will pass through vertex A and vertex D.

We are going to write the equation of a straight line passing through points A and D. We know that, the of a straight line if we have two points say $\left( {{x}_{1}},{{y}_{1}} \right)\And \left( {{x}_{2}},{{y}_{2}} \right)$ is equal to:
$y-{{y}_{1}}=\left( \dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \right)\left( x-{{x}_{1}} \right)$
The two points through which the angle bisector of angle A is passing through are A (-1, 11) and D (3, -5) so using the above equation of straight line through two points we can write the equation passing through A and D.
$\begin{aligned} & y-11=\left( \dfrac{-5-11}{3-\left( -1 \right)} \right)\left( x-\left( -1 \right) \right) \\\ & \Rightarrow y-11=\dfrac{-16}{4}\left( x+1 \right) \\\ \end{aligned}$
In the above equation, 16 will be divided by 4 by 4 times so the above equation will look like:
$\begin{aligned} & y-11=-4\left( x+1 \right) \\\ & \Rightarrow y-11=-4x-4 \\\ & \Rightarrow 4x+y=7 \\\ \end{aligned}$
From the above, we have found the equation of angle bisector of angle A is $4x+y=7$.

So, the correct answer is “Option (b)”.

Note: Instead of writing the whole equation of angle bisector of angle A, we can find the slope of the line which is passing through A and D and then check which option has the same slope as that of angle bisector.
The slope of line passing through two points say $\left( {{x}_{1}},{{y}_{1}} \right)\And \left( {{x}_{2}},{{y}_{2}} \right)$ is equal to:
$m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
The two points we have are A (-1, 11) and D (3, -5) so using the above slope formula to find the slope of angle bisector of angle A.
$\begin{aligned} & m=\dfrac{-5-11}{3-\left( -1 \right)} \\\ & \Rightarrow m=\dfrac{-16}{4}=-4 \\\ \end{aligned}$
Hence, we get the slope of the angle bisector as -4.
Now, checking the slope of option (a) $4x-y=7$ .
We know that the slope for a straight line say $ax+by+c=0$ is equal to the negative of coefficient of x divided by coefficient of y we get,
$-\dfrac{a}{b}$
Using this formula for the slope of the straight line we can find the slope for $4x-y=7$.
$-\dfrac{4}{-1}=4$
This slope is not matching so this is an incorrect option.
Checking the slope for option (b) $4x+y=7$ which is equal to the negative of coefficient of x divided by coefficient of y we get,
$-\dfrac{4}{1}=-4$
This slope is matching with the slope that we have solved so this is the correct option.