4. The equation of the perpendicular bisector is y =
x -
5. The equation of the perpendicular bisector is y = - 2x + 16
6. The equation of the perpendicular bisector is y =
x +
Step-by-step 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
![-\frac{1}{m}](/tpl/images/0540/9217/e0bc7.png)
(reciprocal m and change its sign)The perpendicular bisector of a line means another line perpendicular to it and intersect it in its mid-pointThe formula of the slope of a line is
![m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}](/tpl/images/0540/9217/6fb82.png)
The mid point of a segment whose end points are
![(x_{1},y_{1})](/tpl/images/0540/9217/c7e02.png)
and
![(x_{2},y_{2})](/tpl/images/0540/9217/00b5e.png)
is
![(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})](/tpl/images/0540/9217/4f405.png)
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
∵
= 7 and
= 4
∵
= 2 and
= 6
∴ ![m=\frac{6-2}{4-7}=\frac{4}{-3}](/tpl/images/0540/9217/666a0.png)
- Reciprocal it and change its sign to find the slope of the ⊥ line
∴ The slope of the perpendicular line = ![\frac{3}{4}](/tpl/images/0540/9217/f11e3.png)
- Use the rule of the mid-point to find the mid-point of the line
∴ The mid-point = ![(\frac{7+4}{2},\frac{2+6}{2})](/tpl/images/0540/9217/01711.png)
∴ The mid-point = ![(\frac{11}{2},\frac{8}{2})=(\frac{11}{2},4)](/tpl/images/0540/9217/5b4aa.png)
- Substitute the value of the slope in the form of the equation
∵ y =
x + b
- To find b substitute x and y in the equation by the coordinates
  of the mid-point
∵ 4 =
×
+ b
∴ 4 =
+ b
- Subtract Â
from both sides
∴
= b
∴ y =
x - ![\frac{1}{8}](/tpl/images/0540/9217/7440a.png)
∴ The equation of the perpendicular bisector is y =
x -
5.
∵ The line passes through (8 , 5) and (4 , 3)
- Use the formula of the slope to find its slope
∵
= 8 and
= 4
∵
= 5 and
= 3
∴ ![m=\frac{3-5}{4-8}=\frac{-2}{-4}=\frac{1}{2}](/tpl/images/0540/9217/32d3b.png)
- 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 = ![(\frac{8+4}{2},\frac{5+3}{2})](/tpl/images/0540/9217/85fe9.png)
∴ The mid-point = ![(\frac{12}{2},\frac{8}{2})](/tpl/images/0540/9217/7ffcc.png)
∴ 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
∵
= 6 and
= 0
∵
= 1 and
= -3
∴ ![m=\frac{-3-1}{0-6}=\frac{-4}{-6}=\frac{2}{3}](/tpl/images/0540/9217/d6bfa.png)
- Reciprocal it and change its sign to find the slope of the ⊥ line
∴ The slope of the perpendicular line = ![-\frac{3}{2}](/tpl/images/0540/9217/b3738.png)
- Use the rule of the mid-point to find the mid-point of the line
∴ The mid-point = ![(\frac{6+0}{2},\frac{1+-3}{2})](/tpl/images/0540/9217/d086f.png)
∴ The mid-point = ![(\frac{6}{2},\frac{-2}{2})](/tpl/images/0540/9217/76361.png)
∴ The mid-point = (3 , -1)
- Substitute the value of the slope in the form of the equation
∵ y =
x + b
- To find b substitute x and y in the equation by the coordinates
  of the mid-point
∵ -1 =
× 3 + b
∴ -1 =
+ b
- Add Â
 to both sides
∴
= b
∴ y =
x + ![\frac{7}{2}](/tpl/images/0540/9217/447a9.png)
∴ The equation of the perpendicular bisector is y =
x +