Question

1. endpoint: (-8, -10), midpoint: (10, -7)
2. endpoint: (-2, 7), midpoint: (12, -10)

Explanation:

Step1: Recall mid - point formula

The mid - point formula between 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 $(x_1,y_1)$ and the unknown endpoint be $(x_2,y_2)$ and the mid - point be $(m_x,m_y)$. Then $m_x=\frac{x_1 + x_2}{2}$ and $m_y=\frac{y_1 + y_2}{2}$.

Step2: Solve for $x_2$ in the first problem

For the first problem with $(x_1,y_1)=(-8,-10)$ and $(m_x,m_y)=(10,-7)$. Using $m_x=\frac{x_1 + x_2}{2}$, we substitute the values: $10=\frac{-8 + x_2}{2}$. Multiply both sides by 2: $20=-8 + x_2$. Then add 8 to both sides: $x_2=20 + 8=28$.

Step3: Solve for $y_2$ in the first problem

Using $m_y=\frac{y_1 + y_2}{2}$, substitute the values: $-7=\frac{-10 + y_2}{2}$. Multiply both sides by 2: $-14=-10 + y_2$. Then add 10 to both sides: $y_2=-14 + 10=-4$.

Step4: Solve for $x_2$ in the second problem

For the second problem with $(x_1,y_1)=(-2,7)$ and $(m_x,m_y)=(12,-10)$. Using $m_x=\frac{x_1 + x_2}{2}$, we substitute the values: $12=\frac{-2 + x_2}{2}$. Multiply both sides by 2: $24=-2 + x_2$. Then add 2 to both sides: $x_2=24+2 = 26$.

Step5: Solve for $y_2$ in the second problem

Using $m_y=\frac{y_1 + y_2}{2}$, substitute the values: $-10=\frac{7 + y_2}{2}$. Multiply both sides by 2: $-20=7 + y_2$. Then subtract 7 from both sides: $y_2=-20 - 7=-27$.

Answer:

For the first problem, the other endpoint is $(28,-4)$. For the second problem, the other endpoint is $(26,-27)$.