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Answer:
The required equation is:
[tex]y = -\frac{3}{2}x+9[/tex]
Step-by-step explanation:
To find the equation of a line, the slope and y-intercept is required.
The slope can be found by finding the slope of given line segment. A the perpendicular bisector of a line is perpendicular to the given line, the product of their slopes will be -1 and it will pass through the mid-point of given line segment.
Given points are:
[tex](x_1,y_1) = (8,10)\\(x_2,y_2) = (-4,2)[/tex]
We will find the slope of given line segment first
[tex]m = \frac{y_2-y_1}{x_2-x_1}\\= \frac{2-10}{-4-8}\\=\frac{-8}{-12}\\=\frac{2}{3}[/tex]
Let m_1 be the slope of perpendicular bisector then,
[tex]m.m_1 = -1\\\frac{2}{3}.m_1 = -1\\m_1 = \frac{-3}{2}[/tex]
Now the mid-point
[tex](x,y) = (\frac{x_1+x_2}{2} , \frac{y_1+y_2}{2})\\= (\frac{8-4}{2} , \frac{10+2}{2})\\=(\frac{4}{2}, \frac{12}{2})\\=(2,6)[/tex]
We have to find equation of a line with slope -3/2 passing through (2,6)
The equation of line in slope-intercept form is given by:
[tex]y = m_1x+b[/tex]
Putting the value of slope
[tex]y= -\frac{3}{2}x+b[/tex]
Putting the point (2,6) to find the y-intercept
[tex]6 = -\frac{3}{2}(2)+b\\6 = -3+b\\b = 6+3 =9[/tex]
The equation is:
[tex]y = -\frac{3}{2}x+9[/tex]
The equation of the perpendicular bisection that passes through the point (2,6) and has a slope -3/2 is [tex]\rm (y-6)=-\dfrac{3}{2}(x-2)[/tex] and this can be determined by using the point-slope form of the line.
Given :
Endpoints -- (8,10) and (-4,2)
The following steps can be used in order to determine the equation of the perpendicular bisector:
Step 1 - First, determine the slope of the line that passes through points (8,10) and (-4,2).
[tex]\rm m =\dfrac{2-10}{-4-8}[/tex]
[tex]\rm m = \dfrac{8}{12}[/tex]
[tex]\rm m = \dfrac{2}{3}[/tex]
Step 2 - So, the slope of the perpendicular bisector is given by:
[tex]\rm m'm = -1[/tex]
[tex]\rm m' = -\dfrac{3}{2}[/tex]
Step 3 - Now, determine the midpoint of the line that passes through the points (8,10) and (-4,2).
[tex]\rm x = \dfrac{8-4}{2}=2[/tex]
[tex]\rm y = \dfrac{10+2}{2}=6[/tex]
Step 4 - So, the equation of the perpendicular bisection that passes through the point (2,6) and has a slope -3/2 is:
[tex]\rm (y-6)=-\dfrac{3}{2}(x-2)[/tex]
For more information, refer to the link given below:
https://brainly.com/question/11897796
