Answer
See the explanation below.
Work Step by Step
Let $\left( {{x}_{1}},{{y}_{1}} \right)$ be a point and $y=mx+c$ be a line. The target is to find the equation of a line that is perpendicular to the line $y=mx+c$ and passes through the point $\left( {{x}_{1}},{{y}_{1}} \right)$.
The slope of the line $y=mx+c$ is $m$.
The slope of the perpendicular line is the negative reciprocal of $-\frac{1}{m}$.
Thus, the slope of the perpendicular line is ${{m}_{1}}=-\frac{1}{m}$.
Use the point slope formula $y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$ ;
Substitute the value of the slope of the line $-\frac{1}{m}$ and point $\left( {{x}_{1}},{{y}_{1}} \right)$ in the equation $y-{{y}_{1}}={{m}_{1}}\left( x-{{x}_{1}} \right)$.
Thus, the equation of the line perpendicular to $y=mx+c$ is
$y-{{y}_{1}}=-\frac{1}{m}\left( x-{{x}_{1}} \right)$
