Question

Find an equation for the perpendicular bisector of the line segment whose endpoints are -7,-2 and 5,4

Answer 1

The equation of the perpendicular bisector of the line segment whose endpoints are (-7 , -2) and (5 , 4) is y = -2x - 1

Step-by-step explanation:

Let us revise some rules

• The product of the slopes of the perpendicular line is -1, that means if the slope of one line is m, then the slope of the other is $\frac{-1}{m}$
• The formula of the slope of a line whose endpoints are $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is $m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
• The mid-point of a line whose endpoints are $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is $(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$

∵ A line has endpoints (-7 , -2) and (5 , 4)

∴ $x_{1}$ = -7 and $x_{2}$ = 5

∴ $y_{1}$ = -2 and $y_{2}$ = 4

- Use the formula of the slope up to find the slope of the line

∴ $m=\frac{4-(-2)}{5-(-7)}=\frac{4+2}{5+7}=\frac{6}{12}=\frac{1}{2}$

To find the slope of the perpendicular line to the given line reciprocal it and change its sign

∵ The slope of the given line = $\frac{1}{2}$

∴ The slope of the perpendicular line = -2

∵ The perpendicular line is a bisector of the given line

- That means the perpendicular line intersect the given line

   at its midpoint

∵ The mid point of the given line = $(\frac{-7+5}{2},\frac{-2+4}{2})$

∴ The mid point of the given line = $(\frac{-2}{2},\frac{2}{2})$

∴ The mid point of the given line = (-1 , 1)

Now we wand to find the equation of the line whose slope is -2 and passes through point (-1 , 1)

∵ The form of the equation is y = mx + b, where m is the slope

   and b is the y-intercept

∵ m = -2

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

∴ y = -2x + b

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

   of a point on the line

∵ Point (-1 , 1) lies on the line

∴ x = -1 and y = 1

∵ 1 = -2(-1) + b

∴ 1 = 2 + b

- Subtract 2 from both sides

∴ -1 = b

- Substitute the value of b in the equation

∴ y = -2x + (-1)

∴ y = -2x - 1

The equation of the perpendicular bisector of the line segment whose endpoints are (-7 , -2) and (5 , 4) is y = -2x - 1

Learn more:

You can learn more about the equations of the perpendicular lines in brainly.com/question/9527422

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Answer 2

Answer:

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Step-by-step explanation: