The perpendicular bisector of the line segment AB is the line that passes through the midpoint of AB and is perpendicular to AB.

Find the equation of the perpendicular bisector of the line segment joining the points (1,2) and (7,4). Enter your answer in the form "y = mx + b."

The product of the slopes of the perpendicular lines is -1, that means one of them is additive and multiplicative inverse to the other. If the slope of one is m, then the slope of the other is [tex]-\frac{1}{m}[/tex]
The midpoint of a segment whose endpoints are [tex](x_{1},y_{1})[/tex] and [tex](x_{2},y_{2})[/tex] is [tex](\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})[/tex]
The slope of a line that passes through points [tex](x_{1},y_{1})[/tex] and [tex](x_{2},y_{2})[/tex] is [tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]

∵ A line passes through points (1 , 2) and (7 , 4)

∴ [tex]x_{1}[/tex] = 1 and [tex]x_{2}[/tex] = 7

∴ [tex]y_{1}[/tex] = 2 and [tex]y_{2}[/tex] = 4

∴ The slope of the line = [tex]\frac{4-2}{7-1}=\frac{2}{6}=\frac{1}{3}[/tex]

To find the slope of the perpendicular bisector of this line reciprocal

its slope and change its sign

∵ The reciprocal of [tex]\frac{1}{3}[/tex] is 3

∴ The slope of the perpendicular bisector = -3

Let us find the mid point of the line

∵ [tex]x_{1}[/tex] = 1 and [tex]x_{2}[/tex] = 7

∵ [tex]y_{1}[/tex] = 2 and [tex]y_{2}[/tex] = 4

∴ The midpoint = [tex](\frac{1+7}{2},\frac{2+4}{2})[/tex] = (4 , 3)

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

the line and b is the y-intercept

∵ m = -3

∴ The equation of the line is y = -3x + b

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

of the midpoint

∵ The coordinates of the midpoint are x = 4 and y = 3

∴ 3 = -3(4) + b

∴ 3 = -12 + b

- Add 12 to both sides

∴ 15 = b

∴ the equation of the perpendicular bisector is y = -3x + 15

The equation of the perpendicular bisector of the line segment

joining the points (1 , 2) and (7 , 4) is y = -3x + 15

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You can learn more about the perpendicular lines in brainly.com/question/2601054

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