The equation of the line passing through point (4,2) and perpendicular to line AB, with a gradient of 3, is y = 3x - 10.
To find the equation of the line passing through point (4,2) and perpendicular to AB, you can use the following steps:
Find the gradient of line AB by using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of points A and B. In this case, m = (1 - 3) / (7 - 1) = -1/3.
Find the gradient of a line perpendicular to AB by using the fact that the product of the gradients of two perpendicular lines is -1. Therefore, if m1 is the gradient of AB and m2 is the gradient of the perpendicular line, then m1 * m2 = -1. Solving for m2 gives m2 = -1 / m1. In this case, m2 = -1 / (-1/3) = 3.
Find the equation of the line passing through point (4,2) and perpendicular to AB by using the point-slope form of the equation of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. In this case, y - 2 = 3(x - 4). Simplifying this gives y = 3x - 10.
Therefore, the equation of the line passing through point (4,2) and perpendicular to AB is y = 3x - 10.