Final answer:
The equation of the perpendicular bisector of the line segment with endpoints at (-3, -1) and (5, 3) is y = -2x + 3, which is derived by calculating the midpoint, finding the negative reciprocal of the original slope, and applying the slope-intercept form.
Step-by-step explanation:
To write the equation of the perpendicular bisector of the given line segment with endpoints at (-3, -1) and (5, 3), we need to follow these steps:
- Calculate the midpoint of the line segment to find the point through which the perpendicular bisector will pass. The midpoint (M) is obtained by averaging the x-coordinates and the y-coordinates of the endpoints: M = ((-3 + 5) / 2, (-1 + 3) / 2) = (1, 1).
- Find the slope of the original line segment. The slope, m, between two points (x1, y1) and (x2, y2) is given by (y2 - y1) / (x2 - x1). For our points, m = (3 - (-1)) / (5 - (-3)) = 4 / 8 = 0.5.
- The slope of the perpendicular bisector is the negative reciprocal of the original slope. Hence, the slope of the bisector, mbisector, is -2 because -2 is the negative reciprocal of 0.5.
- Use the slope-intercept form of the line equation, y = mx + b, to find the equation of the perpendicular bisector. With the slope mbisector and midpoint M(1,1), we substitute into the equation: 1 = (-2)(1) + b, thus b = 3.
- The equation of the perpendicular bisector is y = -2x + 3.