The equation of the perpendicular bisector is y = [tex]-\frac{7}{2}[/tex] x + 2
Step-by-step explanation:
Let us revise the relation between the slopes of perpendicular lines
The mid point of a segment whose endpoints are [tex](x_{1},y_{1})[/tex] and [tex](x_{2},y_{2})[/tex] is [tex](\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})[/tex]
The perpendicular bisector of a line is the line that intersect it in its mid-point and formed 4 right angles
∵ The end point of a given line are (9 , -3) and (-5 , -7)
∴ [tex]x_{1}=9[/tex] and [tex]x_{2}=-5[/tex]
∴ [tex]y_{1}=-3[/tex] and [tex]y_{2}=-7[/tex]
- Find the slope of the line by using the rule of the slope [tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
∵ [tex]m=\frac{-7-(-3)}{-5-9}=\frac{-7+3}{-14}=\frac{-4}{-14}=\frac{2}{7}[/tex]
∴ The slope of the given line is [tex]\frac{2}{7}[/tex]
To find the slope of the perpendicular bisector of it reciprocal it and change its sign
∴ The slope of the perpendicular bisector = [tex]-\frac{7}{2}[/tex]
∵ The form of the linear equation is y = mx + b, where m is the
slope and b is the y-intercept
- Substitute the value of m in the equation
∴ The equation of the perpendicular bisector is y = [tex]-\frac{7}{2}[/tex] x + b
To find b substitute x and y in the equation by a point on the line
∵ The perpendicular bisector of the given line intersect it at
its midpoint
- Find the mid-point of the given line busing the rule above
∵ [tex]x_{1}=9[/tex] and [tex]x_{2}=-5[/tex]
∵ [tex]y_{1}=-3[/tex] and [tex]y_{2}=-7[/tex]
∴ The mid-point of the given line = [tex](\frac{9+(-5)}{2},\frac{-3+(-7)}{2})=(\frac{4}{2},\frac{-10}{2})=(2,-5)[/tex]
Point (2 , -5) is also lies on the perpendicular line
∴ x = 2 and y = -5
- Substitute them in the equation
∵ -5 = [tex]-\frac{7}{2}[/tex] (2) + b
∴ -5 = -7 + b
- Add 7 to both sides
∴ 2 = b
- Substitute the value of b in the equation
∴ The equation of the perpendicular bisector is y = [tex]-\frac{7}{2}[/tex] x + 2
The equation of the perpendicular bisector is y = [tex]-\frac{7}{2}[/tex] x + 2
Learn more:
You can learn more about the linear equation in brainly.com/question/11223427
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