Chapter 1.1, Problem 53E

To determine

Determine the other endpoint $(x_2,y_2)$ if one of the endpoint is $(x_1,y_1)$ and mid point is $(x_m,y_m)$ .

Answer to Problem 53E

  $(2x_m-x_1,2y_m-y_1)$

Explanation of Solution

Given:

Endpoint: $(x_1,y_1)$

Mid point: $(x_m,y_m)$

Formula used:

The midpoint of a line segment passing through two points $(m_1,n_1)\text{and}(m_2,n_2)$ is given by

  $(m_m,n_m)=\left[\frac{m_1+m_2}{2},\frac{n_1+n_2}{2}\right]$

Calculation:

To find other endpoint, substitute given endpoint and midpoint in above formula.

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

By comparing on both sides,

  $\begin{array}{cc}\begin{array}{l}\Rightarrow x_m=\frac{x_1+x_2}{2}\\ \Rightarrow 2x_m=x_1+x_2\\ \Rightarrow 2x_m-x_1=x_2\\ \Rightarrow x_2=2x_m-x_1\end{array}& \begin{array}{l}\Rightarrow y_m=\frac{y_1+y_2}{2}\\ \Rightarrow 2y_m=y_1+y_2\\ \Rightarrow 2y_m-y_1=y_2\\ \Rightarrow y_2=2y_m-y_1\end{array}\end{array}$

So, the other endpoint is $(x_2,y_2)=(2x_m-x_1,2y_m-y_1).$

Conclusion:

Therefore, point $(2x_m-x_1,2y_m-y_1)$ is other endpoint.