Question

the mid - point of $overline{ab}$ is $m(0,2)$. if the coordinates of $a$ are $(1,3)$, 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 $x_1 = 1,y_1 = 3$ and $M(0,2)$.

Step2: Solve for $x_2$

We know that $\frac{x_1 + x_2}{2}=0$. Substitute $x_1 = 1$ into the equation: $\frac{1 + x_2}{2}=0$. Multiply both sides by 2: $1 + x_2=0$. Then subtract 1 from both sides: $x_2=-1$.

Step3: Solve for $y_2$

We know that $\frac{y_1 + y_2}{2}=2$. Substitute $y_1 = 3$ into the equation: $\frac{3 + y_2}{2}=2$. Multiply both sides by 2: $3 + y_2 = 4$. Then subtract 3 from both sides: $y_2 = 1$.

Answer:

$(-1,1)$