Chapter 1.1, Problem 48E

a.

To determine

Find the coordinates of the endpoint of a line segment using the Midpoint Formula.

Answer to Problem 48E

  $\left(7,0\right)$

Explanation of Solution

Given information:

To find the coordinates of the endpoint of a line segment when the coordinates of the other endpoint and midpoint are, respectively,

  $\left(1,2\right),\left(4,-1\right)$

Calculation:

The one end of line segment $\left(x_1,y_1\right)$ and other end $\left(x_2,y_2\right)$ having midpoint $\left(x_m,y_m\right)$

The coordinate value of $\left(x_1,y_1\right)$ and midpoint $\left(x_m,y_m\right)$ are given.

Now find coordinate value of $\left(x_2,y_2\right)$ by using midpoint formula,

  $\begin{array}{l}{x}_m=\frac{\left(x_1+x_2\right)}{2}\\ y_m=\frac{\left(y_1+y_2\right)}{2}\end{array}$

  $\begin{array}{l}{x}_m=\frac{\left(x_1+x_2\right)}{2}\\ 2x_m=\left(x_1+x_2\right)\\ \boxed{x_2=2x_m-x_1}\end{array}$

Now substitute the given values,

  $\begin{array}{l}{x}_2=2\left(4\right)-1\\ =8-1\\ x_2=7\end{array}$

  $\begin{array}{l}{y}_m=\frac{\left(y_1+y_2\right)}{2}\\ 2y_m=\left(y_1+y_2\right)\\ \boxed{y_2=2y_m-y_1}\end{array}$

Now substitute the given values,

  $\begin{array}{l}{y}_2=2\left(-1\right)-\left(-2\right)\\ y_2=0\end{array}$

Hence the coordinates of other end of line segment $\left(7,0\right)$

b.

To determine

Find the coordinates of the endpoint of a line segment using the Midpoint Formula.

Answer to Problem 48E

  $\left(9,-3\right)$

Explanation of Solution

Given information:

To find the coordinates of the endpoint of a line segment when the coordinates of the other endpoint and midpoint are, respectively,

  $\left(-5,11\right),\left(2,4\right)$

Calculation:

The one end of line segment $\left(x_1,y_1\right)$ and other end $\left(x_2,y_2\right)$ having midpoint $\left(x_m,y_m\right)$

The coordinate value of $\left(x_1,y_1\right)$ and midpoint $\left(x_m,y_m\right)$ are given.

Now find coordinate value of $\left(x_2,y_2\right)$ by using midpoint formula,

  $\begin{array}{l}{x}_m=\frac{\left(x_1+x_2\right)}{2}\\ y_m=\frac{\left(y_1+y_2\right)}{2}\end{array}$

  $\begin{array}{l}{x}_m=\frac{\left(x_1+x_2\right)}{2}\\ 2x_m=\left(x_1+x_2\right)\\ \boxed{x_2=2x_m-x_1}\end{array}$

Now substitute the given values,

  $\begin{array}{l}{x}_2=2\left(2\right)-\left(-5\right)\\ =4+5\\ x_2=9\end{array}$

  $\begin{array}{l}{y}_m=\frac{\left(y_1+y_2\right)}{2}\\ 2y_m=\left(y_1+y_2\right)\\ \boxed{y_2=2y_m-y_1}\end{array}$

Now substitute the given values,

  $\begin{array}{l}{y}_2=2\left(4\right)-11\\ y_2=-3\end{array}$

Hence the coordinates of other end of line segment $\left(9,-3\right)$