Question

the vertices of triangle ABC are A(-2,3), B(0,-3), and C(4,1). prove, by means of coordinate geometry, that the median to side BC is also the altitude to side BC.

Answer 1

A median is a line joining the a vertex to the mid-point of the opposite side.
If the median is also an altitude to side BC, then the median should be perpendicular to BC.

Let D be the mid-point of BC. The coordinates of D should be rather obvious, being the mid-point of points B and C.

Compare the slopes of lines AC and lines BC. If the product of the two slopes equal -1, the lines are perpendicular, hence AD is an altitude to side BC.