Chapter 4.6, Problem 50STP

a.

To determine

Find the given coordinate value using the table.

Answer to Problem 50STP

The coordinate of $C\text{'}$ is $\left(-17,-1\right)$ .

Explanation of Solution

Given:

The given table is

  

The given point is $C\text{'}$ .

  $A\text{'}B\text{'}C\text{'}D\text{'}$ is the translation of the quadrilateral $ABCD$ .

Calculation:

Write a matrix equation.

  $\left[\begin{array}{cccc}2& 9& -14& -6\\ 5& -15& -9& 1\end{array}\right]+\left[\begin{array}{cccc}x& x& x& x\\ y& y& y& y\end{array}\right]=\left[\begin{array}{cccc}-1& a& p& -9\\ 13& b& q& 9\end{array}\right]$

  $\Rightarrow \left[\begin{array}{cccc}2+x& 9+x& -14+x& -6+x\\ 5+y& -15+y& -9+y& 1+y\end{array}\right]=\left[\begin{array}{cccc}-1& a& p& -9\\ 13& b& q& 9\end{array}\right]$

The matrices are equal, so corresponding elements are equal. Use the elements from the first row in each matrix.

  $\begin{array}{l}2+x=-1\\ \Rightarrow 2+x-2=-1-2\\ \Rightarrow x=-3\end{array}$

  $\begin{array}{l}5+y=13\\ \Rightarrow 5+y-5=13-5\\ \Rightarrow y=8\end{array}$

Coordinate of $C\text{'}$ is

  $\begin{array}{l}-14+x=p\\ \Rightarrow -14+\left(-3\right)=p\\ \Rightarrow p=-17\end{array}$

  $\begin{array}{l}-9+y=q\\ \Rightarrow -9+8=q\\ \Rightarrow q=-1\end{array}$

Hence the Coordinate of $C\text{'}$ is $\left(-17,-1\right)$ .

a.

To determine

Find the given coordinate value using the table.

Answer to Problem 50STP

The coordinate of $B\text{'}$ is $\left(6,-7\right)$ .

Explanation of Solution

Given:

The given table is

  

The given point is $C\text{'}$ .

  $A\text{'}B\text{'}C\text{'}D\text{'}$ is the translation of the quadrilateral $ABCD$ .

Calculation:

Write a matrix equation.

  $\left[\begin{array}{cccc}2& 9& -14& -6\\ 5& -15& -9& 1\end{array}\right]+\left[\begin{array}{cccc}x& x& x& x\\ y& y& y& y\end{array}\right]=\left[\begin{array}{cccc}-1& a& p& -9\\ 13& b& q& 9\end{array}\right]$

  $\Rightarrow \left[\begin{array}{cccc}2+x& 9+x& -14+x& -6+x\\ 5+y& -15+y& -9+y& 1+y\end{array}\right]=\left[\begin{array}{cccc}-1& a& p& -9\\ 13& b& q& 9\end{array}\right]$

The matrices are equal, so corresponding elements are equal. Use the elements from the first row in each matrix.

  $\begin{array}{l}2+x=-1\\ \Rightarrow 2+x-2=-1-2\\ \Rightarrow x=-3\end{array}$

  $\begin{array}{l}5+y=13\\ \Rightarrow 5+y-5=13-5\\ \Rightarrow y=8\end{array}$

Coordinate of $B\text{'}$ is

  $\begin{array}{l}9+x=a\\ \Rightarrow 9+\left(-3\right)=a\\ \Rightarrow a=6\end{array}$

  $\begin{array}{l}-15+y=b\\ \Rightarrow -15+8=b\\ \Rightarrow b=-7\end{array}$

Hence the Coordinate of $B\text{'}$ is $\left(6,-7\right)$ .