How to find the midpoint of a line?

Question

How to find the other end of a line with only midpoint and one other end.

Ex:

M is the midpoint of AB. The coordinates of M are (-3, 6) and the coordinates of B are (2, 9). What are the  coordinates of A?  

Answer 1

The midpoint of a line segment is equidistant from the ends of the line. If you subtract the coordinates of an endpoint from the coordinates of the midpoint, you'll find the distance in the x-direction and in the y-direction between the endpoint and the midpoint. You can then add this distance to the midpoint to find the other endpoint. Mathematically, it's M - B = (-3, 6) - (2, 9) = (-3 - 2, 6 - 9) = (-5, -3)

Now add this to the midpoint to find the coordinates of A: (-3, 6) + (-5, -3) = (-3 - 5, 6 -3) = (-8, 3)

You can check your work by taking the average of A and B and seeing if you get M:

[A + B]/2 = [(-8, 3) + (2, 9)] / 2 = (-8 + 2, 3 + 9) / 2 = (-6, 12) / 2 = (-3, 6). Because this matches the coordinates of the midpoint, you know your solution is correct.