Question

the mid - point of $overline{jk}$ is $m(6,3)$. one endpoint is $j(14,9)$. find the coordinates of endpoint $k$.

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 $J(x_1,y_1)=(4,9)$ and $K(x_2,y_2)$, and $M(6,3)$.

Step2: Solve for $x_2$

We know that $\frac{x_1 + x_2}{2}=x_M$. Substituting $x_1 = 4$ and $x_M=6$ into the formula: $\frac{4 + x_2}{2}=6$. Multiply both sides by 2: $4 + x_2=12$. Then subtract 4 from both sides: $x_2=12 - 4=8$.

Step3: Solve for $y_2$

We know that $\frac{y_1 + y_2}{2}=y_M$. Substituting $y_1 = 9$ and $y_M = 3$ into the formula: $\frac{9 + y_2}{2}=3$. Multiply both sides by 2: $9 + y_2=6$. Then subtract 9 from both sides: $y_2=6 - 9=-3$.

Answer:

$(8,-3)$