Chapter EP, Problem 4.4.1EP

(a)

To determine

To Find: The coordinates of the image ABCD in the vertex matrix.

Answer to Problem 4.4.1EP

The vertex matrix is $ABCD=\left[\begin{array}{cccc}1& -2& -4& 2\\ 1& 3& -1& -3\end{array}\right]$ .

Explanation of Solution

Given:

The vertices of the quadrilateral ABCD are $A\left(1,1\right),B\left(-2,3\right),C\left(-4,-1\right),D\left(2,-3\right)$ .

Calculation:

Consider the given matrix is $A\left(1,1\right),B\left(-2,3\right),C\left(-4,-1\right),D\left(2,-3\right)$ .

Then, the coordinate of the quadrilateral in the matrix form is,

  $ABCD=\left[\begin{array}{cccc}1& -2& -4& 2\\ 1& 3& -1& -3\end{array}\right]$

(b)

To determine

To Find: The coordinates of $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ .

Answer to Problem 4.4.1EP

The required vertices are $A^{\prime }\left(2,2\right)B^{\prime }\left(-4,6\right)C^{\prime }\left(-8,-2\right)D^{\prime }\left(4,-6\right)$ .

Explanation of Solution

Consider the vertex matrix is $ABCD=\left[\begin{array}{cccc}1& -2& -4& 2\\ 1& 3& -1& -3\end{array}\right]$ .

Since, the dilation factor is 2 so the coordinate of the dilated matrix is,

  $2\left[\begin{array}{cccc}1& -2& -4& 2\\ 1& 3& -1& -3\end{array}\right]=\left[\begin{array}{cccc}2& -4& -8& 4\\ 2& 6& -2& -6\end{array}\right]$

The vertices of the dilated matrix is,

  $A^{\prime }\left(2,2\right)B^{\prime }\left(-4,6\right)C^{\prime }\left(-8,-2\right)D^{\prime }\left(4,-6\right)$

(c)

To determine

To Find: the graph of $ABCD$ and $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ .

Answer to Problem 4.4.1EP

The required graph is shown in Figure 1.

Explanation of Solution

The vertices of the dilated quadrilateral is $A^{\prime }\left(2,2\right)B^{\prime }\left(-4,6\right)C^{\prime }\left(-8,-2\right)D^{\prime }\left(4,-6\right)$ and the vertices of quadrilateral is $A\left(1,1\right),B\left(-2,3\right),C\left(-4,-1\right),D\left(2,-3\right)$ .

Then the graph is shown in Figure 1.

  

Figure 1