Question

1. a line segment has one endpoint at (-5,7) and a midpoint at (-1,-2). determine the other endpoint and the length of the line segment. (4 marks)

Explanation:

Step1: Find the other endpoint

Let the endpoints of the line - segment be $(x_1,y_1)=(-5,7)$ and $(x_2,y_2)$ and the mid - point $(x_m,y_m)=(-1,-2)$.
The mid - point formula is $x_m=\frac{x_1 + x_2}{2}$ and $y_m=\frac{y_1 + y_2}{2}$.
For the x - coordinate:
$-1=\frac{-5 + x_2}{2}$
Multiply both sides by 2: $-2=-5 + x_2$.
Add 5 to both sides: $x_2=-2 + 5=3$.
For the y - coordinate:
$-2=\frac{7 + y_2}{2}$
Multiply both sides by 2: $-4=7 + y_2$.
Subtract 7 from both sides: $y_2=-4 - 7=-11$.
So the other endpoint is $(3,-11)$.

Step2: Calculate the length of the line segment

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
Here, $(x_1,y_1)=(-5,7)$ and $(x_2,y_2)=(3,-11)$.
$d=\sqrt{(3-(-5))^2+(-11 - 7)^2}=\sqrt{(3 + 5)^2+(-18)^2}=\sqrt{8^2+(-18)^2}=\sqrt{64 + 324}=\sqrt{388}=2\sqrt{97}$.

Answer:

The other endpoint is $(3,-11)$ and the length of the line segment is $2\sqrt{97}$.