answer:To calculate the length of the line segment AB, we can use the distance formula: Distance = √[(x2 - x1)^2 + (y2 - y1)^2] Given that point A is (4, -3) and point B is (-1, 9), we can substitute their coordinates into the formula: Distance = √[(-1 - 4)^2 + (9 - (-3))^2] = √[(-5)^2 + (12)^2] = √[25 + 144] = √169 = 13 Therefore, the length of line segment AB is 13 units.

answer:To find the midpoint of line segment AB, we can use the midpoint formula: Midpoint x-coordinate = (x1 + x2)/2 Midpoint y-coordinate = (y1 + y2)/2 Given that point A is (4, -3) and point B is (-1, 9), we can substitute their coordinates into the formula: Midpoint x-coordinate = (4 + (-1))/2 = 3/2 = 1.5 Midpoint y-coordinate = (-3 + 9)/2 = 6/2 = 3 Therefore, the midpoint of line segment AB is (1.5, 3).

answer:To find the equation of the line l that is perpendicular to the line x - 3y + 15 = 0 and passes through the midpoint of AB, we need to find the slope of the line x - 3y + 15 = 0 and then determine the slope of the line perpendicular to it. First, let's rearrange the equation x - 3y + 15 = 0 into slope-intercept form: x - 3y + 15 = 0 -3y = -x - 15 y = (1/3)x + 5 From the equation in slope-intercept form, we can see that the slope of the line is 1/3. The slope of a line perpendicular to another line is the negative reciprocal of its slope. Therefore, the slope of line l is -(3/1) = -3. Now, let's find the midpoint of AB. Given that point A is (-2, 6) and point B is (3, -8), we can use the midpoint formula: Midpoint x-coordinate = (x1 + x2)/2 = (-2 + 3)/2 = 1/2 Midpoint y-coordinate = (y1 + y2)/2 = (6 + (-8))/2 = -1 Therefore, the midpoint of AB is (1/2, -1). Since we know the slope of line l and a point it passes through (the midpoint of AB), we can use the point-slope form of a line to find the equation of line l: y - y1 = m(x - x1) where (x1, y1) is a point on the line and m is the slope. Substituting the values: y - (-1) = -3(x - 1/2) y + 1 = -3x + 3/2 y = -3x + 3/2 - 1 y = -3x + 3/2 - 2/2 y = -3x + 1/2 Finally, rearranging the equation to the required form ax + by + c = 0, where a, b, and c are integers: 3x + y - 1/2 = 0 6x + 2y - 1 = 0 6x + 2y - 1 = 0 (multiply both sides by 2 to get rid of fractions) 6x + 2y - 1 = 0 Therefore, the equation of line l is 6x + 2y - 1 = 0.

answer:In the equation of line l, which is 6x + 2y - 1 = 0, we can identify the values of a, b, and c. The equation is already in the form ax + by + c = 0, where a, b, and c are the coefficients. Comparing the given equation to the required form: 6x + 2y - 1 = 0 We can deduce that: a = 6 b = 2 c = -1 Thus, in the equation 6x + 2y - 1 = 0, the values of a, b, and c are 6, 2, and -1, respectively.