Final answer:
The gradient of AB is -1. The mid-point of AB is (3, 2). The equation of the line perpendicular to AB is y = x - 1.
Step-by-step explanation:
The gradient of a line is the ratio of the vertical change (rise) to the horizontal change (run) between two points. To find the gradient of AB, we need to calculate the difference in the y-coordinates and divide it by the difference in the x-coordinates. In this case, the y-coordinates are 4 and 0, and the x-coordinates are 2 and 4. So the gradient of AB is (0-4)/(4-2) = -2/2 = -1.
The mid-point of AB is the point that lies exactly halfway between A and B. To find it, we need to calculate the average of the x-coordinates and the average of the y-coordinates. In this case, the x-coordinates are 2 and 4, and the y-coordinates are 4 and 0. So the mid-point of AB is ((2+4)/2, (4+0)/2) = (3, 2).
The line perpendicular to AB will have a gradient that is the negative reciprocal of the gradient of AB. Since the gradient of AB is -1, the gradient of the perpendicular line will be 1. The equation of a line with gradient 1 passing through the mid-point of AB can be found using the point-slope form of a linear equation. Using the mid-point (3, 2), the equation of the line perpendicular to AB is y - 2 = 1(x - 3), which simplifies to y = x - 1.
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