Question

the midpoint of $overline{ab}$ is $m(3,2)$. if the coordinates of $a$ are $(8,7)$, 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 $x_1 = 8,y_1 = 7$ and the mid - point $M$ has coordinates $(3,2)$.

Step2: Solve for $x_2$

We know that $\frac{x_1 + x_2}{2}=3$. Substitute $x_1 = 8$ into the equation: $\frac{8 + x_2}{2}=3$. Multiply both sides by 2: $8 + x_2=6$. Then subtract 8 from both sides: $x_2=6 - 8=-2$.

Step3: Solve for $y_2$

We know that $\frac{y_1 + y_2}{2}=2$. Substitute $y_1 = 7$ into the equation: $\frac{7 + y_2}{2}=2$. Multiply both sides by 2: $7 + y_2 = 4$. Then subtract 7 from both sides: $y_2=4 - 7=-3$.

Answer:

$(-2,-3)$