Question

6. A rhombus ABCD has vertices A(2. -I).
B(3, 1), C(5, 2) and D(4, 0).
(a) Calculate the image of the rhombus ABCD
under a glide reflection G equivalent to
5
a translation T =
()
and a reflection in the
5
line y = x.
(b) Draw a graph to show the glide reflection.

Answer 1

### 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.