The equation of the line bisecting perpendicularly the segment joining the points \[\left( { - 4,{\rm{ }}6} \right)\] and \[\left( {8,{\rm{ }}8} \right)\] is
A. \[6x + y - 19 = 0\]
B. \[y = 7\]
C. \[6x + 2y - 19 = 0\]
D. \[x + 2y - 7 = 0\]

Hint: A perpendicular line that cuts through the middle of the line connecting the two locations is referred to as a perpendicular bisector. The equation of the line perpendicular to the line formed by the supplied two points can be found by first locating the midpoint of the given points and then using this point.

Formula Used: The equation of a line with slope m running through the point \[\left( {{x_1},{y_1}} \right)\] is:
\[ \Rightarrow \left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)\]

Complete step by step solution: Let the required line segment be \[AB\]
We have been given in the question that \[AB\] bisects the line segment joining the points \[( - 4,6)\]\[(8,8)\]and it means that \[AB\] passes through the midpoint of those two points.
Midpoint of \[( - 4,6)\] and \[(8,8)\] is:
\[ \Rightarrow \left( {\dfrac{{ - 4 + 8}}{2},\dfrac{{6 + 8}}{2}} \right)\]
\[ \Rightarrow (2,7)\]
The points \[( - 4,6)\]\[(8,8)\]are perpendicular to the line\[AB\]joining them and this means that the slope of \[AB\] will be the inverse of the slope of the line joining the points \[( - 4,6)\]and \[(8,8)\]
Now, we have to determine the slope of the line joining the points \[( - 4,6)\]and \[(8,8)\]
We know that the slope of the line with two points will be,
\[ = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
On substituting the corresponding value in the above formula, we get
\[ = \dfrac{{8 - 6}}{{8 + 4}}\]
Add/Subtract in numerator and denominator,
\[ = \dfrac{2}{{12}}\]
On simplification, we get
\[ = \dfrac{1}{6}\]
Now, let us determine the equation of the line.
We already knew that \[AB\] has slope of \[ - 6\] and passes through the point \[(2,7)\]
Therefore, the above statement can be written in the equation format is,
\[y - {y_1} = m(x - x)\]
Now, we have to substitute the value form the above statement, we get
\[y - 7 = - 6(x - 2)\]
On simplifying the above equation, we get the equation in standard form as,
\[6x + y - 19 = 0\]

Option ‘C’ is correct

Note: We used point slope form to determine the equation of line in the aforementioned issue.
The equation of the line if it passes through the two known points \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\] is:
\[ \Rightarrow \left( {y - {y_1}} \right) = \left( {\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}} \right)\left( {x - {x_1}} \right)\]
There are numerous additional methods to determine the equation of line. Depending on the information at hand, we can employ several techniques.