Question

The midpoint of AB is M(5,1). If the coordinates of A are (3,6), what are thecoordinates of B?

Answer 1

We have a segment AB of which we know the coordinates of A(3,6) and the midpoint M(5,1).

We have to find the coordinates of B.

We know that the coordinates of the midpoint M are the average of the coordinates of the endpoints A and B, so we can write:

$\begin{gathered} x_M=\frac{x_A+x_B}{2} \\ 2\cdot x_M=x_A+x_B \\ x_B=2x_M-x_A \end{gathered}$

Now we have the x-coordinate of B in function of the x-coordinates of A and M.

The same can be calculated for the y-coordinate:

$y_B=2y_M-y_A$

Then, we can replace and calculate:

$\begin{gathered} x_B=2x_M-x_A \\ x_B=2\cdot5-3 \\ x_B=10-3 \\ x_B=7 \end{gathered}$$\begin{gathered} y_B=2y_M-y_A \\ y_B=2\cdot1-6 \\ y_B=2-6 \\ y_B=-4 \end{gathered}$

Then, the coordinates of B are (7,-4).

Answer: B = (7,-4)