Chapter 11.1, Problem 86E

To determine

Find the coordinates of the endpoint of a line segment.

Answer to Problem 86E

  $\boxed{x_2=1,y_2=-8,z_2-3}$

Explanation of Solution

Given information:

Find the coordinates of the endpoint of a line segment when the coordinates of the other endpoint and the midpoint are $\left(3,0,2,\right)$ and $\left(5,8,7\right)$ respectively.

Calculation:

Consider the first end points of line segment.

  $\left[x_1,y_1,z_1\right]$

The second end points of line segment.

  $\left[x_2,y_2,z_2\right]$

The mid points of the line segment.

  $\left[x_m,y_m,z_m\right]$

The midpoint of line segment joining two points is,

  $\begin{array}{l}M=\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\\ x_m=\frac{x_1+x_2}{2}\\ x_2=2x_m-x_1\end{array}$

  $\begin{array}{l}{y}_m=\frac{y_1+y_2}{2}\\ y_2=2y_m-y_1\end{array}$

  $\begin{array}{l}{z}_m=\frac{z_1+z_2}{2}\\ z_2=2z_m-z_1\end{array}$

So the second end points of line segment are,

  $\begin{array}{l}{x}_2=2x_m-x_1\\ y_2=2y_m-y_1\\ z_2=2z_m-z_1\end{array}$

Consider the given points, and put the points in above expression.

  $\begin{array}{l}{x}_2=2\times 3-5=1\\ y_2=2\times 0-8=-8\\ z_2=2\times 2-7=-3\end{array}$

Hence, the coordinates of the endpoint of a line segment are $\boxed{x_2=1,y_2=-8,z_2-3}$ .