Question

1. endpoint: (10, 12), mid - point: (6, 9)

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 the given endpoint be $(x_1,y_1)=(10,12)$ and the mid - point be $(x_m,y_m)=(6,9)$. We want to find the other endpoint $(x_2,y_2)$.
For the x - coordinate:
$\frac{x_1 + x_2}{2}=x_m$. Substitute $x_1 = 10$ and $x_m = 6$ into the formula:
$\frac{10 + x_2}{2}=6$.

Step2: Solve for $x_2$

Multiply both sides of the equation $\frac{10 + x_2}{2}=6$ by 2: $10 + x_2=12$. Then subtract 10 from both sides: $x_2=12 - 10=2$.
For the y - coordinate:
$\frac{y_1 + y_2}{2}=y_m$. Substitute $y_1 = 12$ and $y_m = 9$ into the formula:
$\frac{12 + y_2}{2}=9$.

Step3: Solve for $y_2$

Multiply both sides of the equation $\frac{12 + y_2}{2}=9$ by 2: $12 + y_2 = 18$. Then subtract 12 from both sides: $y_2=18 - 12 = 6$.

Answer:

$(2,6)$