Answer:
[tex]y = -2\, x - 6[/tex].
Step-by-step explanation:
Calculate the gradient of line [tex]\sf {AB}[/tex]. For a non-vertical line that goes through [tex](x_0,\, y_0)[/tex] and [tex](x_1,\, y_1)[/tex] where [tex]x_{0} \ne x_{1}[/tex], the gradient would be:
[tex]\displaystyle \text{$\frac{y_{1} - y_{0}}{x_{1} - x_{0}}$ given that $x_{0} \ne x_{1}$}[/tex].
For line [tex]\sf AB[/tex], the two points are [tex]{\sf A}\, (-6,\, -4)[/tex] and [tex]{\sf B}\, (2,\, 0)[/tex]. Hence, the gradient of line [tex]\sf AB\![/tex] would be:
[tex]\begin{aligned} \frac{0 - (-4)}{2 - (-6)} = \frac{1}{2}\end{aligned}[/tex].
The perpendicular bisector of a line segment ([tex]\sf AB[/tex] in this question) is perpendicular to that line segment.
In a cartesian plane, the gradients of two lines perpendicular to one another would be inverse reciprocals. In other words, the product of these two gradients would be [tex](-1)[/tex].
Hence, if [tex]m_{1}[/tex] represents the gradient of [tex]{\sf AB}\!\![/tex], and [tex]m_{2}[/tex] represents the gradient of the perpendicular bisector, then [tex]m_{1} \cdot m_{2} = -1[/tex].
Since the gradient of [tex]{\sf AB}[/tex] is [tex](1/2)[/tex], the gradient of its perpendicular bisector would be:
[tex]\begin{aligned}m_{2} &= \frac{-1}{m_{1}} \\ &= \frac{-1}{1/2} \\ &= -2\end{aligned}[/tex].
The perpendicular bisector of a line segment ([tex]{\sf AB}[/tex] in this question) goes through the midpoint of that line segment.
Apply the midpoint formula to find the midpoint of segment [tex]{\sf AB}[/tex].
If the endpoints of a line segment are [tex](x_{0},\, y_{0})[/tex] and [tex](x_{1},\, y_{1})[/tex], the midpoint of that line segment would be:
[tex]\begin{aligned} \left(\frac{x_0 + x_1}{2},\, \frac{y_{0} + y_{1}}{2}\right)\end{aligned}[/tex].
The two endpoints of segment [tex]{\sf AB}[/tex] are [tex]{\sf A}\, (-6,\, -4)[/tex] and [tex]{\sf B}\, (2,\, 0)[/tex]. The midpoint of segment [tex]{\sf AB}\![/tex] would be:
[tex]\begin{aligned} \left(\frac{-6 + 2}{2},\, \frac{-4 + 0}{2}\right) &= (-2,\, -2)\end{aligned}[/tex].
Find the equation of this perpendicular bisector in the point-slope form.
Consider a non-vertical line of slope [tex]m[/tex]. If this line goes through the point [tex](x_{0},\, y_{0})[/tex], the point-slope form equation of this line would be:
[tex]y - y_{0} = m\, (x - x_{0})[/tex].
The slope of the perpendicular bisector in this question is [tex](-2)[/tex]. Besides, this line goes through the point [tex](-2,\, -2)[/tex]. Henec, the point-slope equation of this line would be:
[tex]y - (-2) = (-2)\, (x - (-2))[/tex].
[tex]y + 2 = -2\, (x + 2)[/tex].
All the choices in this question are in the slope-intercept form. Hence, rewrite the point-slope equation [tex]y + 2 = -2\, (x + 2)[/tex] to find the corresponding (equivalent) slope-intercept equation of this line:
[tex]y + 2 = -2\, (x + 2)[/tex].
[tex]y = -2\, x - 6[/tex].
