High school geometry

Question

The given line segment has a midpoint at (3, 1). 

What is the equation, in slope-intercept form, of the perpendicular bisector of the given line segment?

y = x

y = x – 2

y = 3x

y = 3x − 8

Answer 1

The perpendicular bisector of the line segment would contain the point (3,1) and have a slope that is the negative reciprocal of the slope of the line between (2,4) and (4,-2)

that slope is (-2-4)/(4-2) = -6/2 = -3.

so the slope of the perpendicular line would be 1/3

equation of a line is y = mx + b, where m is slope and b is y-intercept.

we have the slope m = 1/3 and coordinates of a point the line passes through (3,1) which we can substitute for x and y, then solve for b:

1 = 1/3(3) + b

b = 0

so equation of the line is y = 1/3x

Answer 2

The perpendicular bisector has a slope that is the negative reciprocal of the "blue line" and it passes through the midpoint (3, 1). The slope of the "blue line" is calculated using the slope equation:

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

The negative reciprocal of -3 (aka -3/1) is 1/3 so the slope of the line we are looking for is 1/3. We can plug that into the point-slope form of a line:

Point-slope form: y - y₁ = m(x - x₁) where "m" is the slope and (x₁, y₁) is a point on the line (note that this (x₁, y₁) doesn't have anything to do with the (x₁, y₁) we used to find the slope. In this case, the point is (3, 1) so x1 = 3 and y1 = 1):

y - y₁ = m(x - x₁)

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

y - 1 = 1/3x - 1

y = 1/3x