Points P and Q are situated at coordinates (5,2) and (-7,8) respectively. Find a) The coordinates of the midpoint M of the line PQ [2 marks] b) The equation of the normal of the line PQ passing through the midpoint M [3 marks]

a) For finding the midpoint M, the point M must be equidistant from P and Q in both the x and y axes. Hence, we consider the x and y axis separately. The midpoint of the x coordinates is essentially a mean of their numerical values so the midpoint of the points in the x axis is (5+(-7))/2 and in the y axis (2+8)/2. When put in a calculator this finds that M has the coordinates (-1,5).b) To find the equation of a line requires the coordinates of a point on the line and the gradient of that line (in this case the normal). The gradient of a line has equation (y2-y1)/(x2-x1). Substituting those values into the equation where x1=5, y1=2, x2=-7 and y2=8, we get (8-2)/(-7-5)=6/(-12)=-0.5The gradient of the normal when multiplied to the gradient of the line is -1. Given that, we can obtain the gradient of the normal as (-1)/(-0.5) =2. This allows for the use of the equation of a straight line rule where:y-y1=m(x-x1)By substituting values in, we can obtain y-(-5)=2(x-(-1)), which simplifies to y=2x+7