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 $x_1 = 3,y_1 = 1$, and $M(1,3)$.

Step2: Solve for $x_2$

We know that $\frac{x_1 + x_2}{2}=1$. Substitute $x_1 = 3$ into the equation: $\frac{3 + x_2}{2}=1$. Multiply both sides by 2: $3 + x_2=2$. Then subtract 3 from both sides: $x_2=2 - 3=-1$.

Step3: Solve for $y_2$

We know that $\frac{y_1 + y_2}{2}=3$. Substitute $y_1 = 1$ into the equation: $\frac{1 + y_2}{2}=3$. Multiply both sides by 2: $1 + y_2 = 6$. Then subtract 1 from both sides: $y_2=6 - 1 = 5$.

Answer:

$(-1,5)$