Chapter 4.4, Problem 41PPS

(a)

To determine

To write: The vertex matrices for $\Delta ABC$ with vertices $A\left(-2,3\right)$ , $B\left(4,0\right)$ and $C\left(2,-5\right)$ , and its image $\Delta A\text{'}B\text{'}C\text{'}$ after reflection in the x -axis.

Answer to Problem 41PPS

The vertex matrix is $\left[\begin{array}{ccc}-2& 4& 2\\ -3& 0& 5\end{array}\right]$ and coordinates of the vertices of the image are $A\text{'}\left(-2,-3\right)$ , $B\text{'}\left(4,0\right)$ and $C\text{'}\left(2,5\right)$ .

Explanation of Solution

Given:

The vertices are $A\left(-2,3\right)$ , $B\left(4,0\right)$ and $C\left(2,-5\right)$ and these are reflected in the x -axis.

Calculation:

Write the vertex matrix as follows.

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

Multiply the vertex matrix by the matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ .

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

Therefore, the coordinates of the vertices of the image are $A\text{'}\left(-2,-3\right)$ , $B\text{'}\left(4,0\right)$ and $C\text{'}\left(2,5\right)$ .

The graph of the image and pre-image is as follows.

(b)

To determine

To graph: The $\Delta ABC$ and $\Delta A\text{'}B\text{'}C\text{'}$ .

Answer to Problem 41PPS

The graph of $\Delta ABC$ and $\Delta A\text{'}B\text{'}C\text{'}$ is shown in figure (1).

Explanation of Solution

Given:

The vertices are $A\left(-2,3\right)$ , $B\left(4,0\right)$ and $C\left(2,-5\right)$ and these are reflected in the x -axis.

Calculation:

The graph of $\Delta ABC$ and $\Delta A\text{'}B\text{'}C\text{'}$ is shown below.

  

Figure (1)

Therefore, the graph of $\Delta ABC$ and $\Delta A\text{'}B\text{'}C\text{'}$ is shown in figure (1).

(c)

To determine

To make: A conjecture about the reflection of $\Delta A\text{'}B\text{'}C\text{'}$ in the x -axis and perform the matrix multiplication to verify in conjecture.

Answer to Problem 41PPS

The conjecture is the reflection of $\Delta A\text{'}B\text{'}C\text{'}$ in the x -axis is $\Delta ABC$ .

Explanation of Solution

Given:

The vertices are $A\left(-2,3\right)$ , $B\left(4,0\right)$ and $C\left(2,-5\right)$ and these are reflected in the x -axis.

Calculation:

The conjecture is the reflection of $\Delta A\text{'}B\text{'}C\text{'}$ in the x -axis is $\Delta ABC$ .

Write the vertex matrix as follows.

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

Multiply the vertex matrix by the matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ .

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

Therefore, the matrix is $\left[\begin{array}{ccc}-2& 4& 2\\ -3& 0& 5\end{array}\right]$ .

The graph of the image and pre-image is as follows.

(d)

To determine

To find: The matrix that could be used to reflect a triangle in the x -axis twice and the way by which the matrix is related to the observation.

Answer to Problem 41PPS

The matrix is identity matrix because reflecting a triangle in the same line twice produces the original figure.

Explanation of Solution

Given:

The vertices are $A\left(-2,3\right)$ , $B\left(4,0\right)$ and $C\left(2,-5\right)$ and these are reflected in the x -axis.

Calculation:

The matrix is identity matrix because reflecting a triangle in the same line twice produces the original figure.