Final answer:
The slopes of segment XY and segment WZ can be found by comparing changes in the y-coordinates and changes in the x-coordinates. The slope of segment XY is -1 and the slope of segment WZ is 1. Since these slopes are opposite reciprocals of each other, option 1 is correct that the segments are perpendicular.
Step-by-step explanation:
The given lines x-y = -3 and x - y = 1 define two segments on the coordinate plane, segment XY and segment WZ. To determine how these segments are related, we can look at their slopes. The slope of a line can be found by comparing the changes in the y-coordinates and the changes in the x-coordinates between two points on the line. For segment XY, let's find two points that lie on the line x-y = -3. We can choose any x-coordinate and then calculate the corresponding y-coordinate using the equation. Let's choose x = 0, then y = -3. So one point on the line is (0, -3). Now, let's choose x = 1, then y = -4. Therefore, another point on the line is (1, -4). The change in y-coordinates is -4 - (-3) = -1 and the change in x-coordinates is 1 - 0 = 1. Therefore, the slope of segment XY is -1/1 = -1. For segment WZ, using the same process with the line x - y = 1, we let x = 0, then y = -1, so a point on the line is (0, -1). If we let x = 1, then y = 0, and another point on the line is (1, 0). The change in y-coordinates is 0 - (-1) = 1 and the change in x-coordinates is 1 - 0 = 1. Therefore, the slope of segment WZ is 1/1 = 1. Comparing the slopes of segment XY (-1) and segment WZ (1), we can see that the slopes are not the same, so the segments are not parallel. But they are opposite reciprocals of each other, since -1 = 1/-1. Therefore, the correct statement is option 1: they have slopes that are opposite reciprocals of 1 and -1 and are, therefore, perpendicular.