We have two straight lines AB and CD. The coordinates of A,B and C are A(1,3), B(5,9) and C(0,8). The point D lies on the line AB and is halfway between points A and B. Is the line CD perpendicular to AB?

First of all we need to find the coordinates of the point D. As D is halfway between the two points A and B, to find the midpoint of a line segment, we add the x coordinates then divide by 2, and add the y coordinates and divide by 2. This gives us D(3,6).To find the gradient of AB, we need to divide the change in the y-coordinate by the change in the x-coordinate. (9-3)/(5-1) =3/2 so 3/2 is the gradient of the line segment AB. As we now know our D coordinates we can work out the gradient of CD. (8-6)/(0-3)=-2/3 which is the gradient of line segment CD.If two lines are perpendicular to one another then: (gradient of AB) x (gradient of CD) = -1We then check this : 3/2 x -2/3 = -1 . So AB and CD are indeed perpendicular.