Chapter 1.1, Problem 54E

a .

To determine

To find: is the endpoint $\left(x_2,y_2\right)$ of the line segment with the given endpoint $\left(x_1,y_1\right)=\left(1,-2\right)$ and midpoint $\left(x_m,y_m\right)=\left(4,-1\right)$ .

Answer to Problem 54E

  $\left(7,0\right)$

Explanation of Solution

Concept Used:

Midpoint Formula: Midpoint $\left(x_m,y_m\right)$ of the line segment joining the two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given by:

  $\left(x_m,y_m\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

In the above midpoint formula substitute $\left(x_1,y_1\right)=\left(1,-2\right)$ and $\left(x_m,y_m\right)=\left(4,-1\right)$ ,

  $\begin{array}{c}\left(x_m,y_m\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ \left(4,-1\right)=\left(\frac{1+x_2}{2},\frac{-2+y_2}{2}\right)\end{array}$

Now, equate the x -coordinates and y -coordinates with each other.

  $\begin{array}{c}\frac{1+x_2}{2}=4\\ 1+x_2=8\\ x_2=8-1\\ x_2=7\end{array}$

And

  $\begin{array}{c}\frac{-2+y_2}{2}=-1\\ -2+y_2=-2\\ y_2=-2+2\\ y_2=0\end{array}$

Conclusion:

Hence, the endpoint $\left(x_2,y_2\right)$ of the line segment is $\left(7,0\right)$ .

b.

To determine

To find: is the endpoint $\left(x_2,y_2\right)$ of the line segment with the given endpoint $\left(x_1,y_1\right)=\left(-5,11\right)$ and midpoint $\left(x_m,y_m\right)=\left(2,4\right)$ .

Answer to Problem 54E

  $\left(9,-3\right)$

Explanation of Solution

Concept Used:

Midpoint Formula: Midpoint $\left(x_m,y_m\right)$ of the line segment joining the two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given by:

  $\left(x_m,y_m\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

In the above midpoint formula substitute $\left(x_1,y_1\right)=\left(-5,11\right)$ and $\left(x_m,y_m\right)=\left(2,4\right)$ ,

  $\begin{array}{c}\left(x_m,y_m\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ \left(2,4\right)=\left(\frac{-5+x_2}{2},\frac{11+y_2}{2}\right)\end{array}$

Now, equate the x -coordinates and y -coordinates with each other.

  $\begin{array}{c}\frac{-5+x_2}{2}=2\\ -5+x_2=4\\ x_2=4+5\\ x_2=9\end{array}$

And

  $\begin{array}{c}\frac{11+y_2}{2}=4\\ 11+y_2=8\\ y_2=8-11\\ y_2=-3\end{array}$

Conclusion:

Hence, the endpoint $\left(x_2,y_2\right)$ of the line segment is $\left(9,-3\right)$ .