Question

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

Step2: Find the x - coordinate of B

We know that $\frac{x_1 + x_2}{2}=3$. Substitute $x_1 = 7$ into the equation: $\frac{7 + x_2}{2}=3$. Multiply both sides by 2: $7 + x_2=6$. Then solve for $x_2$: $x_2=6 - 7=-1$.

Step3: Find the y - coordinate of B

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 solve for $y_2$: $y_2=-4 - 3=-7$.

Answer:

$(-1,-7)$