Answer :
To reflect the given triangle over the line [tex]\( y = -x \)[/tex], we will start by understanding the original triangle's vertices and then performing the necessary transformation.
The original vertices of the triangle are given by the coordinates:
[tex]\[ \left[ \begin{array}{ccc} 0 & 7 & -1 \\ 0 & -3 & -4 \end{array} \right] \][/tex]
These vertices correspond to the points [tex]\( (0,0), (7,-3), \text{and} (-1,-4) \)[/tex].
Reflecting a point over the line [tex]\( y = -x \)[/tex] can be achieved by transforming any point [tex]\( (x, y) \)[/tex] to [tex]\( (-y, -x) \)[/tex]. This reflection can be represented by the transformation matrix:
[tex]\[ \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \][/tex]
Let's apply this transformation to each of the vertices:
1. For the vertex at [tex]\( (0, 0) \)[/tex]:
[tex]\[ \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \][/tex]
So, the reflected vertex is [tex]\( (0, 0) \)[/tex].
2. For the vertex at [tex]\( (7, -3) \)[/tex]:
[tex]\[ \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 7 \\ -3 \end{pmatrix} = \begin{pmatrix} 3 \\ -7 \end{pmatrix} \][/tex]
So, the reflected vertex is [tex]\( (3, -7) \)[/tex].
3. For the vertex at [tex]\( (-1, -4) \)[/tex]:
[tex]\[ \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} -1 \\ -4 \end{pmatrix} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} \][/tex]
So, the reflected vertex is [tex]\( (4, 1) \)[/tex].
Therefore, the coordinates of the reflected triangle are:
[tex]\[ \left[ \begin{array}{ccc} 0 & 3 & 4 \\ 0 & -7 & 1 \end{array} \right] \][/tex]
Thus, after reflecting the triangle over the line [tex]\( y = -x \)[/tex], the new vertices of the triangle are [tex]\( (0, 0), (3, -7), \text{and} (4, 1) \)[/tex].
The original vertices of the triangle are given by the coordinates:
[tex]\[ \left[ \begin{array}{ccc} 0 & 7 & -1 \\ 0 & -3 & -4 \end{array} \right] \][/tex]
These vertices correspond to the points [tex]\( (0,0), (7,-3), \text{and} (-1,-4) \)[/tex].
Reflecting a point over the line [tex]\( y = -x \)[/tex] can be achieved by transforming any point [tex]\( (x, y) \)[/tex] to [tex]\( (-y, -x) \)[/tex]. This reflection can be represented by the transformation matrix:
[tex]\[ \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \][/tex]
Let's apply this transformation to each of the vertices:
1. For the vertex at [tex]\( (0, 0) \)[/tex]:
[tex]\[ \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \][/tex]
So, the reflected vertex is [tex]\( (0, 0) \)[/tex].
2. For the vertex at [tex]\( (7, -3) \)[/tex]:
[tex]\[ \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 7 \\ -3 \end{pmatrix} = \begin{pmatrix} 3 \\ -7 \end{pmatrix} \][/tex]
So, the reflected vertex is [tex]\( (3, -7) \)[/tex].
3. For the vertex at [tex]\( (-1, -4) \)[/tex]:
[tex]\[ \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} -1 \\ -4 \end{pmatrix} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} \][/tex]
So, the reflected vertex is [tex]\( (4, 1) \)[/tex].
Therefore, the coordinates of the reflected triangle are:
[tex]\[ \left[ \begin{array}{ccc} 0 & 3 & 4 \\ 0 & -7 & 1 \end{array} \right] \][/tex]
Thus, after reflecting the triangle over the line [tex]\( y = -x \)[/tex], the new vertices of the triangle are [tex]\( (0, 0), (3, -7), \text{and} (4, 1) \)[/tex].