Answer :
### Part (a): Calculate the image of the rhombus ABCD under the glide reflection
The glide reflection involves two steps:
1. Translation: Translate each vertex of the rhombus by the vector [tex]\( T = \begin{pmatrix} 5 \\ 5 \end{pmatrix} \)[/tex].
2. Reflection: Reflect each translated vertex over the line [tex]\( y = x \)[/tex].
#### Step 1: Translation
First, apply the translation vector to each vertex.
- Vertex [tex]\( A \)[/tex] has coordinates [tex]\( (2, -1) \)[/tex].
[tex]\[ A' = A + T = \begin{pmatrix} 2 \\ -1 \end{pmatrix} + \begin{pmatrix} 5 \\ 5 \end{pmatrix} = \begin{pmatrix} 7 \\ 4 \end{pmatrix} \][/tex]
- Vertex [tex]\( B \)[/tex] has coordinates [tex]\( (3, 1) \)[/tex].
[tex]\[ B' = B + T = \begin{pmatrix} 3 \\ 1 \end{pmatrix} + \begin{pmatrix} 5 \\ 5 \end{pmatrix} = \begin{pmatrix} 8 \\ 6 \end{pmatrix} \][/tex]
- Vertex [tex]\( C \)[/tex] has coordinates [tex]\( (5, 2) \)[/tex].
[tex]\[ C' = C + T = \begin{pmatrix} 5 \\ 2 \end{pmatrix} + \begin{pmatrix} 5 \\ 5 \end{pmatrix} = \begin{pmatrix} 10 \\ 7 \end{pmatrix} \][/tex]
- Vertex [tex]\( D \)[/tex] has coordinates [tex]\( (4, 0) \)[/tex].
[tex]\[ D' = D + T = \begin{pmatrix} 4 \\ 0 \end{pmatrix} + \begin{pmatrix} 5 \\ 5 \end{pmatrix} = \begin{pmatrix} 9 \\ 5 \end{pmatrix} \][/tex]
#### Step 2: Reflection
Next, reflect each translated vertex over the line [tex]\( y = x \)[/tex]. To reflect a point [tex]\( (x, y) \)[/tex] over the line [tex]\( y = x \)[/tex], swap the coordinates to get [tex]\( (y, x) \)[/tex].
- Reflecting [tex]\( A' \)[/tex] which has coordinates [tex]\( (7, 4) \)[/tex]:
[tex]\[ A'' = (4, 7) \][/tex]
- Reflecting [tex]\( B' \)[/tex] which has coordinates [tex]\( (8, 6) \)[/tex]:
[tex]\[ B'' = (6, 8) \][/tex]
- Reflecting [tex]\( C' \)[/tex] which has coordinates [tex]\( (10, 7) \)[/tex]:
[tex]\[ C'' = (7, 10) \][/tex]
- Reflecting [tex]\( D' \)[/tex] which has coordinates [tex]\( (9, 5) \)[/tex]:
[tex]\[ D'' = (5, 9) \][/tex]
### Final Coordinates after Glide Reflection
The final coordinates of the vertices after the glide reflection are:
- [tex]\( A'' (4, 7) \)[/tex]
- [tex]\( B'' (6, 8) \)[/tex]
- [tex]\( C'' (7, 10) \)[/tex]
- [tex]\( D'' (5, 9) \)[/tex]
### Part (b): Draw a graph to show the glide reflection
#### Steps to draw the graph:
1. Original Rhombus: Plot the original vertices [tex]\( A(2, -1) \)[/tex], [tex]\( B(3, 1) \)[/tex], [tex]\( C(5, 2) \)[/tex], [tex]\( D(4, 0) \)[/tex], and connect them to form rhombus ABCD.
2. Translated Rhombus: Plot the translated vertices [tex]\( A'(7, 4) \)[/tex], [tex]\( B'(8, 6) \)[/tex], [tex]\( C'(10, 7) \)[/tex], [tex]\( D'(9, 5) \)[/tex], and connect them to show the translated rhombus.
3. Reflected Rhombus: Plot the final reflected vertices [tex]\( A''(4, 7) \)[/tex], [tex]\( B''(6, 8) \)[/tex], [tex]\( C''(7, 10) \)[/tex], [tex]\( D''(5, 9) \)[/tex], and connect them to show the rhombus after reflection.
4. Line [tex]\( y = x \)[/tex]: Draw the line [tex]\( y = x \)[/tex] on the graph for reference.
By plotting these on a graph, you will visually see the steps of the glide reflection process. This would involve:
1. Showing the original position of the rhombus.
2. Demonstrating the effect of translation by shifting each vertex 5 units right and 5 units up.
3. Illustrating the effect of reflection by swapping the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-coordinates of each translated vertex.
The glide reflection involves two steps:
1. Translation: Translate each vertex of the rhombus by the vector [tex]\( T = \begin{pmatrix} 5 \\ 5 \end{pmatrix} \)[/tex].
2. Reflection: Reflect each translated vertex over the line [tex]\( y = x \)[/tex].
#### Step 1: Translation
First, apply the translation vector to each vertex.
- Vertex [tex]\( A \)[/tex] has coordinates [tex]\( (2, -1) \)[/tex].
[tex]\[ A' = A + T = \begin{pmatrix} 2 \\ -1 \end{pmatrix} + \begin{pmatrix} 5 \\ 5 \end{pmatrix} = \begin{pmatrix} 7 \\ 4 \end{pmatrix} \][/tex]
- Vertex [tex]\( B \)[/tex] has coordinates [tex]\( (3, 1) \)[/tex].
[tex]\[ B' = B + T = \begin{pmatrix} 3 \\ 1 \end{pmatrix} + \begin{pmatrix} 5 \\ 5 \end{pmatrix} = \begin{pmatrix} 8 \\ 6 \end{pmatrix} \][/tex]
- Vertex [tex]\( C \)[/tex] has coordinates [tex]\( (5, 2) \)[/tex].
[tex]\[ C' = C + T = \begin{pmatrix} 5 \\ 2 \end{pmatrix} + \begin{pmatrix} 5 \\ 5 \end{pmatrix} = \begin{pmatrix} 10 \\ 7 \end{pmatrix} \][/tex]
- Vertex [tex]\( D \)[/tex] has coordinates [tex]\( (4, 0) \)[/tex].
[tex]\[ D' = D + T = \begin{pmatrix} 4 \\ 0 \end{pmatrix} + \begin{pmatrix} 5 \\ 5 \end{pmatrix} = \begin{pmatrix} 9 \\ 5 \end{pmatrix} \][/tex]
#### Step 2: Reflection
Next, reflect each translated vertex over the line [tex]\( y = x \)[/tex]. To reflect a point [tex]\( (x, y) \)[/tex] over the line [tex]\( y = x \)[/tex], swap the coordinates to get [tex]\( (y, x) \)[/tex].
- Reflecting [tex]\( A' \)[/tex] which has coordinates [tex]\( (7, 4) \)[/tex]:
[tex]\[ A'' = (4, 7) \][/tex]
- Reflecting [tex]\( B' \)[/tex] which has coordinates [tex]\( (8, 6) \)[/tex]:
[tex]\[ B'' = (6, 8) \][/tex]
- Reflecting [tex]\( C' \)[/tex] which has coordinates [tex]\( (10, 7) \)[/tex]:
[tex]\[ C'' = (7, 10) \][/tex]
- Reflecting [tex]\( D' \)[/tex] which has coordinates [tex]\( (9, 5) \)[/tex]:
[tex]\[ D'' = (5, 9) \][/tex]
### Final Coordinates after Glide Reflection
The final coordinates of the vertices after the glide reflection are:
- [tex]\( A'' (4, 7) \)[/tex]
- [tex]\( B'' (6, 8) \)[/tex]
- [tex]\( C'' (7, 10) \)[/tex]
- [tex]\( D'' (5, 9) \)[/tex]
### Part (b): Draw a graph to show the glide reflection
#### Steps to draw the graph:
1. Original Rhombus: Plot the original vertices [tex]\( A(2, -1) \)[/tex], [tex]\( B(3, 1) \)[/tex], [tex]\( C(5, 2) \)[/tex], [tex]\( D(4, 0) \)[/tex], and connect them to form rhombus ABCD.
2. Translated Rhombus: Plot the translated vertices [tex]\( A'(7, 4) \)[/tex], [tex]\( B'(8, 6) \)[/tex], [tex]\( C'(10, 7) \)[/tex], [tex]\( D'(9, 5) \)[/tex], and connect them to show the translated rhombus.
3. Reflected Rhombus: Plot the final reflected vertices [tex]\( A''(4, 7) \)[/tex], [tex]\( B''(6, 8) \)[/tex], [tex]\( C''(7, 10) \)[/tex], [tex]\( D''(5, 9) \)[/tex], and connect them to show the rhombus after reflection.
4. Line [tex]\( y = x \)[/tex]: Draw the line [tex]\( y = x \)[/tex] on the graph for reference.
By plotting these on a graph, you will visually see the steps of the glide reflection process. This would involve:
1. Showing the original position of the rhombus.
2. Demonstrating the effect of translation by shifting each vertex 5 units right and 5 units up.
3. Illustrating the effect of reflection by swapping the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-coordinates of each translated vertex.
