Question

write an equation for the perpendicular bisector of the line joining the two points. PLEASE do 4,5 and 6

Answer 1

4. The equation of the perpendicular bisector is y =
`(3)/(4)` x -
`(1)/(8)`

5. The equation of the perpendicular bisector is y = - 2x + 16

6. The equation of the perpendicular bisector is y =
`-(3)/(2)` x +
`(7)/(2)`

Explanation:

Lets revise some important rules

• The product of the slopes of the perpendicular lines is -1, that means if the slope of one of them is m, then the slope of the other is
  `-(1)/(m)` (reciprocal m and change its sign)
• The perpendicular bisector of a line means another line perpendicular to it and intersect it in its mid-point
• The formula of the slope of a line is
  `m=(y_(2)-y_(1))/(x_(2)-x_(1))`
• The mid point of a segment whose end points are
  `(x_(1),y_(1))` and
  `(x_(2),y_(2))` is
  `((x_(1)+x_(2))/(2),(y_(1)+y_(2))/(2))`
• The slope-intercept form of the linear equation is y = m x + b, where m is the slope and b is the y-intercept

4.

∵ The line passes through (7 , 2) and (4 , 6)

- Use the formula of the slope to find its slope

∵
`x_(1)` = 7 and
`x_(2)` = 4

∵
`y_(1)` = 2 and
`y_(2)` = 6

∴
`m=(6-2)/(4-7)=(4)/(-3)`

- Reciprocal it and change its sign to find the slope of the ⊥ line

∴ The slope of the perpendicular line =
`(3)/(4)`

- Use the rule of the mid-point to find the mid-point of the line

∴ The mid-point =
`((7+4)/(2),(2+6)/(2))`

∴ The mid-point =
`((11)/(2),(8)/(2))=((11)/(2),4)`

- Substitute the value of the slope in the form of the equation

∵ y =
`(3)/(4)` x + b

- To find b substitute x and y in the equation by the coordinates

of the mid-point

∵ 4 =
`(3)/(4)` ×
`(11)/(2)` + b

∴ 4 =
`(33)/(8)` + b

- Subtract
`(33)/(8)` from both sides

∴
`-(1)/(8)` = b

∴ y =
`(3)/(4)` x -
`(1)/(8)`

∴ The equation of the perpendicular bisector is y =
`(3)/(4)` x -
`(1)/(8)`

5.

∵ The line passes through (8 , 5) and (4 , 3)

- Use the formula of the slope to find its slope

∵
`x_(1)` = 8 and
`x_(2)` = 4

∵
`y_(1)` = 5 and
`y_(2)` = 3

∴
`m=(3-5)/(4-8)=(-2)/(-4)=(1)/(2)`

- Reciprocal it and change its sign to find the slope of the ⊥ line

∴ The slope of the perpendicular line = -2

- Use the rule of the mid-point to find the mid-point of the line

∴ The mid-point =
`((8+4)/(2),(5+3)/(2))`

∴ The mid-point =
`((12)/(2),(8)/(2))`

∴ The mid-point = (6 , 4)

- Substitute the value of the slope in the form of the equation

∵ y = - 2x + b

- To find b substitute x and y in the equation by the coordinates

of the mid-point

∵ 4 = -2 × 6 + b

∴ 4 = -12 + b

- Add 12 to both sides

∴ 16 = b

∴ y = - 2x + 16

∴ The equation of the perpendicular bisector is y = - 2x + 16

6.

∵ The line passes through (6 , 1) and (0 , -3)

- Use the formula of the slope to find its slope

∵
`x_(1)` = 6 and
`x_(2)` = 0

∵
`y_(1)` = 1 and
`y_(2)` = -3

∴
`m=(-3-1)/(0-6)=(-4)/(-6)=(2)/(3)`

- Reciprocal it and change its sign to find the slope of the ⊥ line

∴ The slope of the perpendicular line =
`-(3)/(2)`

- Use the rule of the mid-point to find the mid-point of the line

∴ The mid-point =
`((6+0)/(2),(1+-3)/(2))`

∴ The mid-point =
`((6)/(2),(-2)/(2))`

∴ The mid-point = (3 , -1)

- Substitute the value of the slope in the form of the equation

∵ y =
`-(3)/(2)` x + b

- To find b substitute x and y in the equation by the coordinates

of the mid-point

∵ -1 =
`-(3)/(2)` × 3 + b

∴ -1 =
`-(9)/(2)` + b

- Add
`(9)/(2)` to both sides

∴
`(7)/(2)` = b

∴ y =
`-(3)/(2)` x +
`(7)/(2)`

∴ The equation of the perpendicular bisector is y =
`-(3)/(2)` x +
`(7)/(2)`