The point A has coordinates (3,1) and the point B has coordinates (-21, 11). The point C is the midpoint of AB. \ (i) Find the equation of the line through A that is perpendicular to y = 2x -7. (ii) Find the distance AC.

(i) To find the equation of the line through A that is perpendicular to y=2x-7, we first need to find the slope of the perpendicular line. The slope of y=2x-7 is 2, so the slope of the perpendicular line is -1/2 (since the product of the slopes of perpendicular lines is -1). So the equation of the line through A with slope -1/2 is y-1 = (-1/2)(x-3), or y=(-1/2)x + 7/2. (ii) To find the distance AC, we first need to find the coordinates of point C. Since C is the midpoint of AB, we can use the midpoint formula: C = ((3+(-21))/2, (1+11)/2) = (-9, 6) Now that we know the coordinates of A and C, we can use the distance formula: distance AC = sqrt(((-9-3)^2) + ((6-1)^2)) = sqrt(144+25) = sqrt(169) = 13 So the distance between points A and C is 13 units.