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____ [38]
3 years ago
9

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

Mathematics
1 answer:
myrzilka [38]3 years ago
8 0

Answer:

4. The equation of the perpendicular bisector is y = \frac{3}{4} x - \frac{1}{8}

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

6. The equation of the perpendicular bisector is y = -\frac{3}{2} x + \frac{7}{2}

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} (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=\frac{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 (\frac{x_{1}+x_{2}}{2},\frac{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=\frac{6-2}{4-7}=\frac{4}{-3}

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

∴ The slope of the perpendicular line = \frac{3}{4}

- 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})

∴ The mid-point = (\frac{11}{2},\frac{8}{2})=(\frac{11}{2},4)

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

∵ y = \frac{3}{4} x + b

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

   of the mid-point

∵ 4 = \frac{3}{4} × \frac{11}{2} + b

∴ 4 = \frac{33}{8} + b

- Subtract  \frac{33}{8} from both sides

∴ -\frac{1}{8} = b

∴ y = \frac{3}{4} x - \frac{1}{8}

∴ The equation of the perpendicular bisector is y = \frac{3}{4} x - \frac{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=\frac{3-5}{4-8}=\frac{-2}{-4}=\frac{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 = (\frac{8+4}{2},\frac{5+3}{2})

∴ The mid-point = (\frac{12}{2},\frac{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=\frac{-3-1}{0-6}=\frac{-4}{-6}=\frac{2}{3}

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

∴ The slope of the perpendicular line = -\frac{3}{2}

- 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})

∴ The mid-point = (\frac{6}{2},\frac{-2}{2})

∴ The mid-point = (3 , -1)

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

∵ y = -\frac{3}{2} x + b

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

   of the mid-point

∵ -1 = -\frac{3}{2} × 3 + b

∴ -1 = -\frac{9}{2} + b

- Add  \frac{9}{2}  to both sides

∴ \frac{7}{2} = b

∴ y = -\frac{3}{2} x + \frac{7}{2}

∴ The equation of the perpendicular bisector is y = -\frac{3}{2} x + \frac{7}{2}

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