Chapter 2.4, Problem 18E

(a)

To determine

To draw:

The graph of the given vertices.

Explanation of Solution

Given:

The vertices is

  $F\left(4,1\right),G\left(0,3\right),H\left(2,-1\right)$

Concept used:

To draw the graph of the given a square with vertices

Calculation:

Matrix represent the triangle

  $\left[\begin{array}{ccc}\begin{array}{l}4\\ 1\end{array}& \begin{array}{l}0\\ 3\end{array}& \begin{array}{l}2\\ -1\end{array}\end{array}\right]$

The matrix of the translation

  $\left[\begin{array}{ccc}\begin{array}{l}4\\ 1\end{array}& \begin{array}{l}0\\ 3\end{array}& \begin{array}{l}2\\ -1\end{array}\end{array}\right]+\left[\begin{array}{ccc}\begin{array}{l}-6\\ -2\end{array}& \begin{array}{l}-6\\ -2\end{array}& \begin{array}{l}-6\\ -2\end{array}\end{array}\right]$

Addition of the matrix

  $\left[\begin{array}{ccc}\begin{array}{l}-2\\ -1\end{array}& \begin{array}{l}-6\\ 1\end{array}& \begin{array}{l}-4\\ -3\end{array}\end{array}\right]$

Graph of triangle $FGH\text{and}{F}^{\prime }{G}^{\prime }{H}^{\prime }$

The coordinate of the translation is $\left(-2,1\right),\left(-6,1\right),\left(-4,-3\right)$

(b)

To determine

To draw:

The graph of the given vertices.

Explanation of Solution

Given:

The vertices is

  $F\left(4,1\right),G\left(0,3\right),H\left(2,-1\right)$

Concept used:

To draw the graph of the given a square with vertices

Calculation:

Matrix represent the triangle

  $\left[\begin{array}{ccc}\begin{array}{l}-2\\ -1\end{array}& \begin{array}{l}-6\\ 1\end{array}& \begin{array}{l}-4\\ -3\end{array}\end{array}\right]$

The matrix of the translation

  $\left[\begin{array}{ccc}\begin{array}{l}-2\\ -1\end{array}& \begin{array}{l}-6\\ 1\end{array}& \begin{array}{l}-4\\ -3\end{array}\end{array}\right]+\left[\begin{array}{ccc}\begin{array}{l}1\\ 5\end{array}& \begin{array}{l}1\\ 5\end{array}& \begin{array}{l}1\\ 5\end{array}\end{array}\right]$

Addition of the matrix

  $\left[\begin{array}{ccc}\begin{array}{l}-1\\ 4\end{array}& \begin{array}{l}-5\\ 6\end{array}& \begin{array}{l}-3\\ 2\end{array}\end{array}\right]$

Graph of triangle $FGH\text{and}{F}^{\prime }{G}^{\prime }{H}^{\prime }$

The coordinate of the translation is $\left(-1,-4\right),\left(-5,6\right),\left(-3,2\right)$

(c)

To determine

To compare:

The results of the given part (a) and part (b).

Explanation of Solution

Given:

The vertices is

  $F\left(4,1\right),G\left(0,3\right),H\left(2,-1\right)$

Concept used:

To draw the graph of the given a square with vertices.

Calculation:

Matrix represent the triangle

  $\left[\begin{array}{ccc}\begin{array}{l}4\\ 1\end{array}& \begin{array}{l}0\\ 3\end{array}& \begin{array}{l}2\\ -1\end{array}\end{array}\right]$

The matrix of the translation

  $\left[\begin{array}{ccc}\begin{array}{l}4\\ 1\end{array}& \begin{array}{l}0\\ 3\end{array}& \begin{array}{l}2\\ -1\end{array}\end{array}\right]+\left[\begin{array}{ccc}\begin{array}{l}-6\\ -2\end{array}& \begin{array}{l}-6\\ -2\end{array}& \begin{array}{l}-6\\ -2\end{array}\end{array}\right]+\left[\begin{array}{ccc}\begin{array}{l}1\\ 5\end{array}& \begin{array}{l}1\\ 5\end{array}& \begin{array}{l}1\\ 5\end{array}\end{array}\right]$

Addition of the matrix

  $\begin{array}{l}\left[\begin{array}{ccc}\begin{array}{l}4\\ 1\end{array}& \begin{array}{l}0\\ 3\end{array}& \begin{array}{l}2\\ -1\end{array}\end{array}\right]+\left[\begin{array}{ccc}\begin{array}{l}-5\\ 3\end{array}& \begin{array}{l}-5\\ 3\end{array}& \begin{array}{l}-5\\ 3\end{array}\end{array}\right]\\ \left[\begin{array}{ccc}\begin{array}{l}-1\\ 4\end{array}& \begin{array}{l}-5\\ 6\end{array}& \begin{array}{l}-3\\ 2\end{array}\end{array}\right]\end{array}$

Therefore, a translation $5$ units left and $3$ units up