Question

the midpoint of $overline{cd}$ is $m=(1, 3)$. one endpoint is $c=(-1, 7)$. find the coordinates of the other endpoint, $d$.

Explanation:

Step1: Recall mid - point formula

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Let $C=(-1,7)=(x_1,y_1)$ and $D=(x_2,y_2)$, and $M=(1,3)$.

Step2: Solve for $x_2$

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

Step3: Solve for $y_2$

We know that $\frac{y_1 + y_2}{2}=3$. Substitute $y_1 = 7$ into the equation: $\frac{7 + y_2}{2}=3$. Multiply both sides by 2: $7+y_2=6$. Subtract 7 from both sides: $y_2=6 - 7=-1$.

Answer:

$(3,-1)$