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Find an equation for the perpendicular bisector of the line segment whose endpoints
(5, -3) and (-7, -7).
are

Sagot :

theleangreenbean

Answer:

[tex]y=-3x-8[/tex]

Step-by-step explanation:

Hi there!

What we need to know:

  • Midpoint: [tex](\frac{x_1+x_2}{2} ,\frac{y_1+y_2}{2} )[/tex] where the endpoints are [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]
  • Linear equations are typically organized in slope-intercept form: [tex]y=mx+b[/tex] where m is the slope and b is the y-intercept (the value of y when the line crosses the y-axis)
  • Perpendicular lines always have slopes that are negative reciprocals (ex. 2 and -1/2, 3/4 and -4/3, etc.)

1) Determine the midpoint of the line segment

When two lines bisect each other, they intersect at the middle of each line, or the midpoint.

[tex](\frac{x_1+x_2}{2} ,\frac{y_1+y_2}{2} )[/tex]

Plug in the endpoints (5, -3) and (-7, -7)

[tex](\frac{5+(-7)}{2} ,\frac{-3+(-7)}{2} )\\(\frac{-2}{2} ,\frac{-10}{2} )\\(-1,-5)[/tex]

Therefore, the midpoint of the line segment is (-1,-5).

2) Determine the slope of the line segment

Recall that the slopes of perpendicular lines are negative reciprocals. Doing this will help us determine the slope of the perpendicular bisector.

Slope = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex] where the given points are [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]

Plug in the endpoints (5, -3) and (-7, -7)

[tex]\frac{-7-(-3)}{-7-5}\\\frac{-7+3}{-7-5}\\\frac{-4}{-12}\\\frac{1}{3}[/tex]

Therefore, the slope of the line segment is [tex]\frac{1}{3}[/tex]. The negative reciprocal of [tex]\frac{1}{3}[/tex] is -3, so the slope of the perpendicular is -3. Plug this into [tex]y=mx+b[/tex]:

[tex]y=-3x+b[/tex]

3) Determine the y-intercept of the perpendicular bisector (b)

[tex]y=-3x+b[/tex]

Recall that the midpoint of the line segment is is (-1,-5), and that the perpendicular bisector passes through this point. Plug this point into [tex]y=-3x+b[/tex] and solve for b:

[tex]-5=-3(-1)+b\\-5=3+b[/tex]

Subtract 3 from both sides

[tex]-5-3=3+b-3\\-8=b[/tex]

Therefore, the y-intercept of the line is -8. Plug this back into [tex]y=-3x+b[/tex]:

[tex]y=-3x-8[/tex]

I hope this helps!

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