Question

Write the equation in slope intercept form for the line perpendicular to c(-4,-5) and D(4,9) passing through the midpoint of the line

Answer 1

Slope intercept form of line passing through midpoint of CD and perpendicular to CD is $\Rightarrow y=-\frac{4}{7} x+2$

Solution:

Need to find the slope intercept form for the line perpendicular to C(-4,-5) and D(4,9)

And passing through the midpoints of the line CD.

Lets first calculate slope of CD  

Let say slope of CD be represented by $m_1$

General formula of slope of line passing through points $\left(x_{1}, y_{1}\right) \text { and }\left(x_{2}, y_{2}\right)$ is as follows:

$m=\frac{\left(y_{2}-y_{1}\right)}{\left(x_{2}-x_{1}\right)}$

$\text { In case of line } \mathrm{CD} , x_{1}=-4, \quad y_{1}=-5 \text { and } x_{2}=4, y_{2}=9$

$\text {So slope of line } \mathrm{CD} \text { that is } m_{1}=\frac{(9-(-5))}{(4-(-4))}=\frac{14}{8}=\frac{7}{4}$

Let’s say slope of required line which is perpendicular to CD be $m_2$

As product of slope of the lines perpendicular to each other is -1

=> slope of line CD $\times$ slope of line perpendicular to CD = -1

$\begin{array}{l}{=>m_{1} \times m_{2}=-1} \\\\ {\Rightarrow \frac{7}{4} \times m_{2}=-1} \\\\ {\Rightarrow m_{2}=-\frac{4}{7}}\end{array}$

Now let’s find midpoint of CD

$\text { Midpoint }(x, y) \text { of two points }\left(x_{1}, y_{1}\right) \text { and }\left(x_{2}, y_{2}\right) \text { is given by }$

$x=\frac{x_{2}+x_{1}}{2} \text { and } y=\frac{y_{2}+y_{1}}{2}$

$\text { So in case of line } \mathrm{CD} , x_{1}=-4, y_{1}=-5 \text { and } x_{2}=4, y_{2}=9$

And midpoint of CD will be as follows

$x=\frac{x_{2}+x_{1}}{2}=\frac{4+(-4)}{2}=0 \text { and } y=\frac{y_{2}+y_{1}}{2}=\frac{9-5}{2}=2$

So midpoint of CD is ( 0 , 2 )

As it is given that line whose slope intercept form is required is perpendicular to CD and passing through midpoint of CD , we need equation of line passing through ( 0 , 2 ) and having slope as $m_{2}=-\frac{4}{7}$

Generic equation of line passing through $\left(x_{1}, y_{1}\right)$ and having slope of m is given by

$\left(y-y_{1}\right)=m\left(x-x_{1}\right)$

$\text { In our case } x_{1}=0 \text { and } y_{1}=2 \text { and } m=-\frac{4}{7}$

Substituting the values in generic equation of line we get

$(y-2)=-\frac{4}{7}(x-0)$

As we required final equation in slope intercept form which is y = mx + c, lets rearrange our equation is required form:

$\Rightarrow y=-\frac{4}{7} x+2$

Hence can conclude that slope intercept form of line passing through midpoint of CD and perpendicular to CD is $\Rightarrow y=-\frac{4}{7} x+2$