Question

find the coordinates of the missing endpoint if b is the midpoint of $overline{ac}$. a(1,7),b(-3,1)

Explanation:

Step1: Recall mid - point formula

The mid - point formula for two points $A(x_1,y_1)$ and $C(x_2,y_2)$ with mid - point $B(x_m,y_m)$ is $x_m=\frac{x_1 + x_2}{2}$ and $y_m=\frac{y_1 + y_2}{2}$.
Given $A(1,7)$ and $B(-3,1)$, for the $x$ - coordinate:
$x_m=-3$, $x_1 = 1$. Substitute into $x_m=\frac{x_1 + x_2}{2}$:
$-3=\frac{1 + x_2}{2}$

Step2: Solve for $x_2$

Multiply both sides of the equation $-3=\frac{1 + x_2}{2}$ by 2:
$-3\times2=1 + x_2$
$-6 = 1+x_2$
Subtract 1 from both sides: $x_2=-6 - 1=-7$

Step3: Solve for $y_2$

For the $y$ - coordinate, $y_m = 1$, $y_1=7$. Substitute into $y_m=\frac{y_1 + y_2}{2}$:
$1=\frac{7 + y_2}{2}$
Multiply both sides by 2: $1\times2=7 + y_2$
$2=7 + y_2$
Subtract 7 from both sides: $y_2=2 - 7=-5$

Answer:

$(-7,-5)$