Write an equation for the perpendicular bisector of the line segment joining the two points.
(1,7), (-3,2)
can somebody explain on how to do this?

Points (1, 7)  and  (-3, 2)

Slope for a line between (x₁, y₁) and (x₂, y₂) , m = (y₂ -y₁) / (x₂- x₁)

The slope for the line joining the two points =  (2 - 7) / (-3 - 1) = -5/-4

Slope = 5/4

Hence the perpendicular bisector would have a slope of -1/(5/4) = -4/5

By condition of perpendicularity

For points (1, 7)  and  (-3, 2),

Formula for midpoints for (x₁, y₁) and (x₂, y₂) is ((x₁ +x₂)/2 , (y₁+ y₂)/2)

Midpoint for (1, 7)  and  (-3, 2) = ((1+ -3)/2 , (7+2)/2) = (-2/2, 9/2)

= (-1, 9/2)

Since the slope of perpendicular bisector is -4/5 and passes through the midpoint (-1, 9/2)

Equation  y - y₁ = m (x - x₁)

                y - 9/2 = (-4/5) (x - -1)
           
                y - 9/2 = (-4/5)(x + 1)

               5(y - 9/2) = -4(x + 1)

               5y - 45/2 = -4x - 4

                 5y =  -4x - 4 + 45/2
           
                5y + 4x = 45/2 - 4

                  5y + 4x = 22  1/2  - 4 =  18 1/2

                  5y + 4x =  37/2

                  10y + 8x = 37

The equation of the line to perpendicular bisector is 10y + 8x = 37

Explanation is in the file

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