Question

Show all work and provide necessary descriptions

Answer 1

Let

$P=(P_x,P_y),\quad Q=(Q_x,Q_y)$

If M is the midpoint, the x and y coordinates of M are the average of the x and y coordinates of P and Q:

$M=\left(\dfrac{P_x+Q_x}{2},\ \dfrac{P_y+Q_y}{2}\right)$

We can solve this expression for the coordinates of Q:

$M_x = \dfrac{P_x+Q_x}{2} \implies Q_x = 2M_x-P_x$

$M_y = \dfrac{P_y+Q_y}{2} \implies Q_y = 2M_y-P_y$

Plug in the values for the coordinates of M and P to get

$Q_x = 2M_x-P_x = 2\cdot 5-11 = 10-11=-1$

$Q_y = 2M_y-P_y = 2\cdot (-2) - (-10) = -4+10=6$