Question

a line segment has an endpoint at (1,1) and a mid - point at (1, - 3). what are the coordinates of the other endpoint? (1,5) (1, - 1) (1, - 5) (1, - 7)

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 known endpoint be $(x_1,y_1)=(1,1)$ and the mid - point be $(x_m,y_m)=(1, - 3)$, and the unknown endpoint be $(x_2,y_2)$.

Step2: Solve for $x_2$

We have $\frac{x_1 + x_2}{2}=x_m$. Substituting $x_1 = 1$ and $x_m = 1$ into the formula: $\frac{1+x_2}{2}=1$. Cross - multiply gives $1 + x_2=2$, so $x_2 = 1$.

Step3: Solve for $y_2$

We have $\frac{y_1 + y_2}{2}=y_m$. Substituting $y_1 = 1$ and $y_m=-3$ into the formula: $\frac{1 + y_2}{2}=-3$. Cross - multiply: $1 + y_2=-6$. Then $y_2=-7$.

Answer:

$(1,-7)$