Final answer:
To find the equation of the perpendicular bisector of a segment, we need to find the midpoint of the segment, calculate the slope of the original segment, determine the negative reciprocal of that slope, and use the point-slope form of a linear equation to find the equation of the perpendicular bisector.
Step-by-step explanation:
To find the equation of the perpendicular bisector of a segment, we first need to find the midpoint of the segment. The midpoint can be found by averaging the x-coordinates and y-coordinates of the endpoints. In this case, the midpoint is (4, 8).
Next, we find the slope of the original segment by using the formula: slope = (y2 - y1) / (x2 - x1) = (11 - 5) / (6 - 2) = 6 / 4 = 1.5.
The slope of the perpendicular bisector is the negative reciprocal of the slope of the original segment. So the slope of the perpendicular bisector is -1 / 1.5 = -2/3.
Now that we have the midpoint and the slope of the perpendicular bisector, we can use the point-slope form of a linear equation to find the equation of the perpendicular bisector. The equation is: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. Plugging in the values, we get y - 8 = -2/3(x - 4). Simplifying the equation gives us y = -2/3x + 10/3.