Answer :

mhanifa

Answer:

  • The perpendicular bisector of the line segment whose endpoints are (-3, 2) and (7, 6) is y = - 2.5x + 9

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Given

  • Points (-3, 2) and (7, 6)

To find

  • The equation of the perpendicular bisector of the segment with given endpoints.

Solution

Find the midpoint of the segment and its slope to determine the perpendicular line passing through the midpoint.

The midpoint has coordinates:

  • x = ( - 3 + 7)/2 = 4/2 = 2,
  • y = (2 + 6)/2 = 8/2 = 4.

The slope:

  • m = (6 - 2)/(7 - (-3)) = 4/10 = 2/5

We know perpendicular lines have opposite/reciprocal slopes.

So the slope of the perpendicular bisector is:

  • m = - 1/ (2/5) = - 5/2 = - 2.5

Use the coordinates of the midpoint and point-slope equation to determine the line:

  • y - 4 = -2.5(x - 2)
  • y - 4 = - 2.5x + 5
  • y = - 2.5x + 5 + 4
  • y = - 2.5x + 9
semsee45

Answer:

[tex]y=-\dfrac{5}{2}x+9[/tex]

Step-by-step explanation:

A perpendicular bisector is a line that intersects another line segment perpendicularly and divides it into two equal parts.

Given endpoints of the line segment:

  • (x₁, y₁) = (-3, 2)
  • (x₂, y₂) = (7, 6)

Substitute the given endpoints into the slope formula to find the slope of the line segment:

[tex]\implies \textsf{Slope $m$}=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{6-2}{7-(-3)}=\dfrac{4}{10}=\dfrac{2}{5}[/tex]

If two lines are perpendicular to each other, their slopes are negative reciprocals.  

Therefore, the slope of the perpendicular line is -⁵/₂.

Find the midpoint of the given line segment by substituting the given endpoints into the midpoint formula:

[tex]\implies \textsf{Midpoint}=\left(\dfrac{x_2+x_1}{2},\dfrac{y_2+y_1}{2}\right)[/tex]

[tex]\implies \textsf{Midpoint}=\left(\dfrac{7-3}{2},\dfrac{6+2}{2}\right)[/tex]

[tex]\implies \textsf{Midpoint}=\left(\dfrac{4}{2},\dfrac{8}{2}\right)[/tex]

[tex]\implies \textsf{Midpoint}=\left(2,4\right)[/tex]

To find the equation of the perpendicular bisector, substitute the found slope and midpoint into the point-slope form of a linear equation:

[tex]\implies y-y_1=m(x-x_1)[/tex]

[tex]\implies y-4=-\dfrac{5}{2}(x-2)[/tex]

[tex]\implies y-4=-\dfrac{5}{2}x+5[/tex]

[tex]\implies y=-\dfrac{5}{2}x+9[/tex]

Therefore, the equation for the perpendicular bisector of the line segment whose endpoints are (-3, 2) and (7, 6) is:

[tex]\boxed{y=-\dfrac{5}{2}x+9}[/tex]

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