Question

PLZZZZZZZZZZ HELP I WILL REPORT IF WRONG

Write the equation of the perpendicular bisector of AB¯¯¯¯¯¯¯¯ if A(–6, –4) and B(2, 0).

Group of answer choices

y=−2x−2

y=−2x−6

y=1/2x−6

y=1/2x−2

Answer 1

`y = -2\, x - 6`.

Explanation:

Calculate the gradient of line
`\sf {AB}`. For a non-vertical line that goes through
`(x_0,\, y_0)` and
`(x_1,\, y_1)` where
`x_(0) \\e x_(1)`, the gradient would be:

`\displaystyle \text{$(y_(1) - y_(0))/(x_(1) - x_(0))$ given that $x_(0) \\e x_(1)$}`.

For line
`\sf AB`, the two points are
`{\sf A}\, (-6,\, -4)` and
`{\sf B}\, (2,\, 0)`. Hence, the gradient of line
`\sf AB\!` would be:

`\begin{aligned} (0 - (-4))/(2 - (-6)) = (1)/(2)\end{aligned}`.

The perpendicular bisector of a line segment (
`\sf AB` 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
`(-1)`.

Hence, if
`m_(1)` represents the gradient of
`{\sf AB}\!\!`, and
`m_(2)` represents the gradient of the perpendicular bisector, then
`m_(1) \cdot m_(2) = -1`.

Since the gradient of
`{\sf AB}` is
`(1/2)`, the gradient of its perpendicular bisector would be:

`\begin{aligned}m_(2) &= (-1)/(m_(1)) \\ &= (-1)/(1/2) \\ &= -2\end{aligned}`.

The perpendicular bisector of a line segment (
`{\sf AB}` in this question) goes through the midpoint of that line segment.

Apply the midpoint formula to find the midpoint of segment
`{\sf AB}`.

If the endpoints of a line segment are
`(x_(0),\, y_(0))` and
`(x_(1),\, y_(1))`, the midpoint of that line segment would be:

`\begin{aligned} \left((x_0 + x_1)/(2),\, (y_(0) + y_(1))/(2)\right)\end{aligned}`.

The two endpoints of segment
`{\sf AB}` are
`{\sf A}\, (-6,\, -4)` and
`{\sf B}\, (2,\, 0)`. The midpoint of segment
`{\sf AB}\!` would be:

`\begin{aligned} \left((-6 + 2)/(2),\, (-4 + 0)/(2)\right) &= (-2,\, -2)\end{aligned}`.

Find the equation of this perpendicular bisector in the point-slope form.

Consider a non-vertical line of slope
`m`. If this line goes through the point
`(x_(0),\, y_(0))`, the point-slope form equation of this line would be:

`y - y_(0) = m\, (x - x_(0))`.

The slope of the perpendicular bisector in this question is
`(-2)`. Besides, this line goes through the point
`(-2,\, -2)`. Henec, the point-slope equation of this line would be:

`y - (-2) = (-2)\, (x - (-2))`.

`y + 2 = -2\, (x + 2)`.

All the choices in this question are in the slope-intercept form. Hence, rewrite the point-slope equation
`y + 2 = -2\, (x + 2)` to find the corresponding (equivalent) slope-intercept equation of this line:

`y + 2 = -2\, (x + 2)`.

`y = -2\, x - 6`.