Write an equation of the perpendicular bisector of the segment with the endpoints (8,10) and ( -4,2).
y =

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

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