Answer:
4. The equation of the perpendicular bisector is y = [tex]\frac{3}{4}[/tex] x - [tex]\frac{1}{8}[/tex]
5. The equation of the perpendicular bisector is y = - 2x + 16
6. The equation of the perpendicular bisector is y = [tex]-\frac{3}{2}[/tex] x + [tex]\frac{7}{2}[/tex]
Step-by-step explanation:
Lets revise some important rules
4.
∵ The line passes through (7 , 2) and (4 , 6)
- Use the formula of the slope to find its slope
∵ [tex]x_{1}[/tex] = 7 and [tex]x_{2}[/tex] = 4
∵ [tex]y_{1}[/tex] = 2 and [tex]y_{2}[/tex] = 6
∴ [tex]m=\frac{6-2}{4-7}=\frac{4}{-3}[/tex]
- Reciprocal it and change its sign to find the slope of the ⊥ line
∴ The slope of the perpendicular line = [tex]\frac{3}{4}[/tex]
- Use the rule of the mid-point to find the mid-point of the line
∴ The mid-point = [tex](\frac{7+4}{2},\frac{2+6}{2})[/tex]
∴ The mid-point = [tex](\frac{11}{2},\frac{8}{2})=(\frac{11}{2},4)[/tex]
- Substitute the value of the slope in the form of the equation
∵ y = [tex]\frac{3}{4}[/tex] x + b
- To find b substitute x and y in the equation by the coordinates
of the mid-point
∵ 4 = [tex]\frac{3}{4}[/tex] × [tex]\frac{11}{2}[/tex] + b
∴ 4 = [tex]\frac{33}{8}[/tex] + b
- Subtract [tex]\frac{33}{8}[/tex] from both sides
∴ [tex]-\frac{1}{8}[/tex] = b
∴ y = [tex]\frac{3}{4}[/tex] x - [tex]\frac{1}{8}[/tex]
∴ The equation of the perpendicular bisector is y = [tex]\frac{3}{4}[/tex] x - [tex]\frac{1}{8}[/tex]
5.
∵ The line passes through (8 , 5) and (4 , 3)
- Use the formula of the slope to find its slope
∵ [tex]x_{1}[/tex] = 8 and [tex]x_{2}[/tex] = 4
∵ [tex]y_{1}[/tex] = 5 and [tex]y_{2}[/tex] = 3
∴ [tex]m=\frac{3-5}{4-8}=\frac{-2}{-4}=\frac{1}{2}[/tex]
- 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 = [tex](\frac{8+4}{2},\frac{5+3}{2})[/tex]
∴ The mid-point = [tex](\frac{12}{2},\frac{8}{2})[/tex]
∴ 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
∵ [tex]x_{1}[/tex] = 6 and [tex]x_{2}[/tex] = 0
∵ [tex]y_{1}[/tex] = 1 and [tex]y_{2}[/tex] = -3
∴ [tex]m=\frac{-3-1}{0-6}=\frac{-4}{-6}=\frac{2}{3}[/tex]
- Reciprocal it and change its sign to find the slope of the ⊥ line
∴ The slope of the perpendicular line = [tex]-\frac{3}{2}[/tex]
- Use the rule of the mid-point to find the mid-point of the line
∴ The mid-point = [tex](\frac{6+0}{2},\frac{1+-3}{2})[/tex]
∴ The mid-point = [tex](\frac{6}{2},\frac{-2}{2})[/tex]
∴ The mid-point = (3 , -1)
- Substitute the value of the slope in the form of the equation
∵ y = [tex]-\frac{3}{2}[/tex] x + b
- To find b substitute x and y in the equation by the coordinates
of the mid-point
∵ -1 = [tex]-\frac{3}{2}[/tex] × 3 + b
∴ -1 = [tex]-\frac{9}{2}[/tex] + b
- Add [tex]\frac{9}{2}[/tex] to both sides
∴ [tex]\frac{7}{2}[/tex] = b
∴ y = [tex]-\frac{3}{2}[/tex] x + [tex]\frac{7}{2}[/tex]
∴ The equation of the perpendicular bisector is y = [tex]-\frac{3}{2}[/tex] x + [tex]\frac{7}{2}[/tex]
