Question

determining the equation of a perpendicular bisector
find the equation of the perpendicular bisector of the given segment by following these steps.

1. determine the slope of the given segment.
2. calculate the mid - point of the given segment.

Explanation:

Step1: Find two - point coordinates

Let the two - end points of the line segment be \((x_1,y_1)\) and \((x_2,y_2)\) from the graph. Assume the points are \((- 2,-2)\) and \((2,4)\).

Step2: Calculate the slope of the given segment

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Substitute \(x_1=-2,y_1 = - 2,x_2 = 2,y_2 = 4\) into it. \(m=\frac{4-(-2)}{2-(-2)}=\frac{4 + 2}{2 + 2}=\frac{6}{4}=\frac{3}{2}\).

Step3: Find the slope of the perpendicular bisector

The slope of a line perpendicular to a line with slope \(m\) is \(m'=-\frac{1}{m}\). Since \(m = \frac{3}{2}\), then \(m'=-\frac{2}{3}\).

Step4: Calculate the mid - point of the given segment

The mid - point formula is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Substitute \(x_1=-2,y_1=-2,x_2 = 2,y_2 = 4\) into it. The mid - point is \((\frac{-2 + 2}{2},\frac{-2+4}{2})=(0,1)\).

Step5: Use the point - slope form to find the equation of the perpendicular bisector

The point - slope form is \(y - y_0=m'(x - x_0)\), where \((x_0,y_0)=(0,1)\) and \(m'=-\frac{2}{3}\). So \(y - 1=-\frac{2}{3}(x - 0)\), which simplifies to \(y=-\frac{2}{3}x + 1\).

Answer:

\(y=-\frac{2}{3}x + 1\)