Question

The equation of the line bisecting perpendicularly the segment joining the points $(-4,6)$ and $(8,8)$ is:

• A. $6x+y-19=0$
• B. $y=7$
• C. $6x+2y-19=0$
• D. $x+2y-7=0$

Answer 1

The correct option is C

$6x+2y-19=0$

Step 1: Determine the midpoint of $(-4,6)$and $(8,8)$

Let the required line segment be $AB$.

Given that: $AB$ bisects the line segment joining the points $(-4,6)$and $(8,8)$. It means that $AB$ passes through the midpoint of those two points.

Midpoint of $(-4,6)$and $(8,8)$ is:

$$\begin{gathered} \Rightarrow \left(\frac{-4+8}{2},\frac{6+8}{2}\right) \\ \Rightarrow \left(2,7\right) \end{gathered}$$

Given that: $AB$ is perpendicular to the line joining the points $(-4,6)$and $(8,8)$. This means that the slope of $AB$ will be the inverse of the slope of the line joining the points $(-4,6)$and $(8,8)$.

Step 2: Determine the slope of the line joining the points $(-4,6)$and $(8,8)$:

$$\begin{gathered} =\frac{y_2-y_1}{x_2-x_1} \\ =\frac{8-6}{8+4} \\ =\frac{2}{12} \\ =\frac{1}{6} \end{gathered}$$

Therefore, the slope of $AB$ will be:

$$\begin{gathered} =-\frac{1}{{\displaystyle \frac{1}{6}}} \\ =-6 \end{gathered}$$

Step 3: Determine the equation of the line

We know, $AB$ passes through the point $(2,7)$ and has slope $=-6$.

Therefore, the equation of the line $AB$ is:

$$\begin{gathered} y-y_1=m(x-x_1) \\ \Rightarrow y-7=-6(x-2) \\ \Rightarrow 6x+y-19=0 \end{gathered}$$

Therefore, the correct answer is Option (C).