Question

Triangle ABC has vertices A(9,8), B(-8,0) and C(10,-8)

a) Show clearly that the equation of the median through A is 2y-3x 11=0

b) the line through C with gradient -2 meets the median through A at the point T. Establish the coordinates of T.

Answer 1

The median from A bisects BC. The mid point of BC is the average of the x and y cords: ((10-8)/2, (0-8)/2)=(1,-4).

a) The slope of the line joining this mid point to A is (8-(-4))/(9-1)=12/8=3/2. So we can now write y=(3/2)x+c where c is to be found. The line passes through A(9,8): 8=9(3/2)+c, c=8-27/2=-11/2, y=(3/2)x-11/2 or 2y=3x-11.

b) The line through C has the form y=-2x+c, and we find c by substituting C(10,-8): -8=-2*10+c, c=12, y=-2x+12. The lines meet when (3x-11)/2=-2x+12; 3x-11=-4x+24; 7x=35, x=5 and y=-10+12=2, so T is the point (5,2).