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 \(\( T = \begin{pmatrix} 5 \\ 5 \end{pmatrix} \)\).
2. Reflection: Reflect each translated vertex over the line \(\( y = x \)\).
#### Step 1: Translation
First, apply the translation vector to each vertex.
- Vertex \(\( A \)\) has coordinates \(\( (2, -1) \)\).
\(\[
A' = A + T = \begin{pmatrix} 2 \\ -1 \end{pmatrix} + \begin{pmatrix} 5 \\ 5 \end{pmatrix} = \begin{pmatrix} 7 \\ 4 \end{pmatrix}
\]\)
- Vertex \(\( B \)\) has coordinates \(\( (3, 1) \)\).
\(\[
B' = B + T = \begin{pmatrix} 3 \\ 1 \end{pmatrix} + \begin{pmatrix} 5 \\ 5 \end{pmatrix} = \begin{pmatrix} 8 \\ 6 \end{pmatrix}
\]\)
- Vertex \(\( C \)\) has coordinates \(\( (5, 2) \)\).
\(\[
C' = C + T = \begin{pmatrix} 5 \\ 2 \end{pmatrix} + \begin{pmatrix} 5 \\ 5 \end{pmatrix} = \begin{pmatrix} 10 \\ 7 \end{pmatrix}
\]\)
- Vertex \(\( D \)\) has coordinates \(\( (4, 0) \)\).
\(\[
D' = D + T = \begin{pmatrix} 4 \\ 0 \end{pmatrix} + \begin{pmatrix} 5 \\ 5 \end{pmatrix} = \begin{pmatrix} 9 \\ 5 \end{pmatrix}
\]\)
#### Step 2: Reflection
Next, reflect each translated vertex over the line \(\( y = x \)\). To reflect a point \(\( (x, y) \)\) over the line \(\( y = x \)\), swap the coordinates to get \(\( (y, x) \)\).
- Reflecting \(\( A' \)\) which has coordinates \(\( (7, 4) \)\):
\(\[
A'' = (4, 7)
\]\)
- Reflecting \(\( B' \)\) which has coordinates \(\( (8, 6) \)\):
\(\[
B'' = (6, 8)
\]\)
- Reflecting \(\( C' \)\) which has coordinates \(\( (10, 7) \)\):
\(\[
C'' = (7, 10)
\]\)
- Reflecting \(\( D' \)\) which has coordinates \(\( (9, 5) \)\):
\(\[
D'' = (5, 9)
\]\)
### Final Coordinates after Glide Reflection
The final coordinates of the vertices after the glide reflection are:
- \(\( A'' (4, 7) \)\)
- \(\( B'' (6, 8) \)\)
- \(\( C'' (7, 10) \)\)
- \(\( D'' (5, 9) \)\)
### Part (b): Draw a graph to show the glide reflection
#### Steps to draw the graph:
1. Original Rhombus: Plot the original vertices \(\( A(2, -1) \)\), \(\( B(3, 1) \)\), \(\( C(5, 2) \)\), \(\( D(4, 0) \)\), and connect them to form rhombus ABCD.
2. Translated Rhombus: Plot the translated vertices \(\( A'(7, 4) \)\), \(\( B'(8, 6) \)\), \(\( C'(10, 7) \)\), \(\( D'(9, 5) \)\), and connect them to show the translated rhombus.
3. Reflected Rhombus: Plot the final reflected vertices \(\( A''(4, 7) \)\), \(\( B''(6, 8) \)\), \(\( C''(7, 10) \)\), \(\( D''(5, 9) \)\), and connect them to show the rhombus after reflection.
4. Line \(\( y = x \)\): Draw the line \(\( y = x \)\) 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 \(\( x \)\)- and \(\( y \)\)-coordinates of each translated vertex.
