Question

Find the equation of the line through the midpoint of (2,-5) and (8,-3) that is perpendicular to the line that passes through the two points.

a) y = 2x - 9
b) y = -2x + 1
c) y = -2x - 9
d) y = 2x + 1

Answer 1

The equation of the line through the midpoint of (2, -5) and (8, -3), perpendicular to the line passing through these points, is y = -2x + 1.

Step-by-step explanation:

To find the slope of the line passing through the given points (2, -5) and (8, -3), use the formula
`\(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\)`.

The slope of this line is
`\(\frac{{-3 - (-5)}}{{8 - 2}} = (2)/(6) = (1)/(3)\)`. The negative reciprocal of
`\((1)/(3)\)` is -3, which is the slope of a line perpendicular to the given line.

Now, find the midpoint of the line segment between (2, -5) and (8, -3) using the midpoint formula
`\(\left(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}\right)\)`. The midpoint is (5, -4). Use the point-slope form
`\(y - y_1 = m(x - x_1)\)` with the slope -3 and the midpoint (5, -4) to find the equation of the perpendicular line, resulting in
`\(y = -2x + 1\)`.