How do I find the equation of the perpendicular bisector of the line segment whose endpoints are (-4, 8) and (-6, -2) using the Midpoint Formula?

Question

Maverick Smith · Accepted Answer

#y=-1/5x+2# First, you must find the midpoint of the segment, the formula for which is #((x_1+x_2)/2,(y_1+y_2)/2)#. This gives #(-5, 3)# as the midpoint. This is the point at which the segment will be bisected. Next, since we are finding a perpendicular bisector, we must determine what slope is perpendicular to that of the existing segment. To determine the segment's slope, we use the slope formula #(y_2-y_1)/(x_2-x_1)#, which gives us a slope of #5#. Perpendicular lines have opposite and reciprocal slopes. The opposite reciprocal of #5# is #-1/5#. We now know that the perpendicular travels through the point #(-5,3)# and has a slope of #-1/5#. Solve for the unknown #b# in #y=mx+b#. #3=-1/5(-5)+b=>3=1+b=>2=b# Therefore, the equation of the perpendicular bisector is #color(blue)(y=-1/5x+2)#.

Parker Allen · Answer

undefined To find the equation of the perpendicular bisector of a line segment using the Midpoint Formula, follow these steps:

1. Find the midpoint of the line segment using the Midpoint Formula: 
   Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

2. Determine the slope of the line containing the given endpoints using the formula: 
   Slope = (y₂ - y₁) / (x₂ - x₁)

3. Find the negative reciprocal of the slope found in step 2. This will be the slope of the perpendicular bisector.

4. Use the midpoint found in step 1 and the slope of the perpendicular bisector found in step 3 to write the equation of the perpendicular bisector in point-slope form:
   y - y₁ = m(x - x₁)

Substitute the midpoint coordinates and the slope into the equation to get the final result.