Question

b is the midpoint of $overline{ac}$. a has coordinates $(14, - 5)$, and b has coordinates $(6, - 9)$. find the coordinates of c.

Explanation:

Step1: Recall mid - point formula

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Let $A(x_1,y_1)=(14,-5)$, $B(x_m,y_m)=(6,-9)$ and $C(x_2,y_2)$.

Step2: Solve for the x - coordinate of C

We know that $x_m=\frac{x_1 + x_2}{2}$. Substituting the known values: $6=\frac{14 + x_2}{2}$. Multiply both sides by 2: $12 = 14+x_2$. Then subtract 14 from both sides: $x_2=12 - 14=-2$.

Step3: Solve for the y - coordinate of C

We know that $y_m=\frac{y_1 + y_2}{2}$. Substituting the known values: $-9=\frac{-5 + y_2}{2}$. Multiply both sides by 2: $-18=-5 + y_2$. Then add 5 to both sides: $y_2=-18 + 5=-13$.

Answer:

$(-2,-13)$