equation of the perpendicular bisector to the line segment

Question

Write the equation of the perpendicular bisector to the line segment with endpoints (4,2) and (-2,6).

Given line:

(4,2) & (-2,6)

m =

New line:

m(perpendicular)=

midpoint=

Answer 1

The perpendicular bisector would go through the midpoint of the segment (hence, "bisector") and would have a slope that would be the negative reciprocal of the segment (hence, "perpendicular").

To find the midpoint, use the midpoint formula:

So, for the segment with endpoints (4, 2) and (-2, 6), the midpoint is:

x_m = (4 + -2)/2 = 1 and y_m = (2 + 6)/2 = 4 so the midpoint is (1, 4)

To find the slope, use the slope equation:

So, the slope of the segment with endpoints (4, 2) and (-2, 6) is:

m = (6 - 2)/(-2 - 4) = 4/(-6) = -2/3

Then, the line perpendicular to this would have a slope of 3/2 (the negative reciprocal of -2/3).

So, the equation of the line we are interested in has a slope of 3/2 and goes through the point (1, 4). You can now use the point-slope form of a line to find the equation. Remember that the point-slope form is:

y - y₁ = m(x - x₁) so plug in m = 3/2 and (x₁, y₁) = (1, 4) so:

y - 4 = 3/2(x - 1)

y - 4 = 3/2x - 3/2

y = 3/2x - 3/2 + 4

y = 3/2x + 5/2

Answer 2

Midpoint of given segment: M((4 - 2)/2, (2 + 6)/2) = (1, 4)

Slope of given line m = (6 - 2)/((- 2 - 4) = - 2/3

Slope m₁ of orthogonal bysector m₁ = 3/2

Equation of perpendicular bysector: y = 4 + 3/2(x - 1); y = 3/2x + 5/2