Question

the midpoint of $overline{ab}$ is $m(1,3)$. if the coordinates of $a$ are $(3,1)$, what are the coordinates of $b$?

Explanation:

Step1: Recall mid - point formula

The mid - point formula for two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Here $x_1 = 3,y_1 = 1,M_x=1,M_y = 3$.

Step2: Solve for $x_2$

We know that $\frac{x_1 + x_2}{2}=M_x$. Substituting the values, we have $\frac{3 + x_2}{2}=1$. Cross - multiply: $3+x_2 = 2$. Then $x_2=2 - 3=-1$.

Step3: Solve for $y_2$

We know that $\frac{y_1 + y_2}{2}=M_y$. Substituting the values, we have $\frac{1 + y_2}{2}=3$. Cross - multiply: $1 + y_2=6$. Then $y_2=6 - 1 = 5$.

Answer:

$(-1,5)$