Question

Find an equation for the perpendicular bisector of the line segment whose endpoints

are (-1,5) and (-7, -7).

Answer 1

The equation of the perpendicular bisector is y =
`(-1)/(2)` x - 3

Explanation:

The form of the linear equation is y = m x + b, where

• m is the slope

• b is the y-intercept

The rule of the slope is m =
`(y2-y1)/(x2-x1)` , where

• (x1, y1) and (x2, y2) are two points on the line

• The rule of the mid-point is M = (
  `(x1+x2)/(2),(y1+y2)/(2)`)

• The product of the slopes of the perpendicular lines is -1, that means if the slope of one is m, then the slope of the other is
  `(-1)/(m)` (we reciprocal m and change its sign).

∵ A line passes through points (-1, 5) and (-7, -7)

∴ x1 = -1 ad y1 = 5

∴ x2 = -7 and y2 = -7

→ Use the rule of the slope above to find the slope of the line

∵ m =
`(-7-5)/(-7--1)` =
`(-12)/(-7+1)` =
`(-12)/(-6)` = 2

∴ m = 2

→ Reciprocal the value of m and change its sign to find the slope of

the line perpendicular line

∴ m⊥ =
`(-1)/(2)`

→ Substitute in the form of the equation above

∵ y =
`(-1)/(2)` x + b

∵ The ⊥ line is also the bisector of the given line, find the mid-point

of the given line because it is also lying on the ⊥ line

∵ M = (
`(-1+-7)/(2),(5+-7)/(2)`) = (
`(-8)/(2),(-2)/(2)`) = (-4, -1)

∴ M = (-4, -1)

→ Substitute the coordinates of M in the equation of the ⊥ line above

∵ x = -4 and y = -1

∴ -1 =
`(-1)/(2)` (-4) + b

∴ -1 = 2 + b

→ Subtract 2 from both sides

∴ -3 = b

→ Substitute the value of b in the equation

∴ y =
`(-1)/(2)` x + -3

∴ y =
`(-1)/(2)` x - 3

∴ The equation of the perpendicular bisector is y =
`(-1)/(2)` x - 3