Isometries revisited – Inverse and Composite Transformations in 2D (Senior 3 Mathematics) | Ganzaa

Inverse and Composite Transformations in 2D

Lessons

Senior 3 · Mathematics

Inverse and Composite Transformations in 2D

4 lessons

Lesson 1 of 4

Isometries revisited

25 min

Patterns on a wall

Imigongo is a Rwandan art form, traditionally made by women using cow dung, with bold spiral and geometric designs in black, white and red. Look at how a panel is built: one shape is repeated by sliding it, flipping it or turning it. Those moves are the isometries you studied in Senior 2. This unit asks what happens when you do one move after another, and how to undo a move. First, a quick review with some new tools.

Definition

An isometry is a transformation that keeps every distance: $A^{'} B^{'} = A B$ for all points $A$ and $B$ . So the image is congruent to the object.

An isometry is direct if it keeps the order of the vertices going round (anticlockwise stays anticlockwise), and opposite if it reverses it, as a mirror does.

Notation used in this unit: $T_{\vec{v}}$ is the translation by $\vec{v}$ ; $S_{X}$ the central symmetry (half-turn) about $X$ ; $M_{m}$ the reflection in the line $m$ ; $R_{X, \theta}$ the rotation about $X$ through $\theta$ (anticlockwise if $\theta > 0$ ).

The four isometries at a glance

Isometry	Fixed by	Rule (examples)	Type
Translation $T_{\vec{v}}$ , $\vec{v} = \begin{pmatrix} a \\ b \end{pmatrix}$	a vector	$(x, y) \mapsto (x + a,\; y + b)$	direct
Central symmetry $S_{X}$ , $X (a, b)$	a centre	$(x, y) \mapsto (2a - x,\; 2b - y)$	direct
Reflection $M_{m}$	a mirror line	$x$ -axis: $(x, - y)$ ; $y$ -axis: $(- x, y)$ ; $y = x$ : $(y, x)$ ; $y = - x$ : $(- y, - x)$ ; $x = a$ : $(2a - x,\; y)$ ; $y = b$ : $(x,\; 2b - y)$	opposite
Rotation $R_{O, \theta}$	a centre and an angle	$+90^\circ$ : $(- y, x)$ ; $180^\circ$ : $(- x, - y)$ ; $-90^\circ$ : $(y, - x)$	direct

Worked example 1: translation and central symmetry

Triangle $P (2, 1)$ , $Q (5, 1)$ , $R (5, 3)$ . Find its image (a) under $T_{\vec{v}}$ with $\vec{v} = \begin{pmatrix} -3 \\ 2 \end{pmatrix}$ ; (b) under the central symmetry about $X (1, 0)$ .

Solution (a) Add $(- 3, 2)$ : $P^{'} (- 1, 3)$ , $Q^{'} (2, 3)$ , $R^{'} (2, 5)$ .

(b) $(x, y) \mapsto (2 \times 1 - x,\; 2 \times 0 - y) = (2 - x,\; -y)$ : $P^{''} (0, - 1)$ , $Q^{''} (- 3, - 1)$ , $R^{''} (- 3, - 3)$ . Check: $X$ is the midpoint of $P P^{''}$ : $\left(\frac{2 + 0}{2}, \frac{1 - 1}{2}\right) = (1, 0)$ . ✓

Worked example 2: two reflections of single points

Reflect $P (2, 1)$ in the line $x = 3$ , and $Q (5, 1)$ in the line $y = - x$ .

Solution In $x = 3$ : $(2 \times 3 - 2,\; 1) = (4, 1)$ ; $P$ is 1 unit left of the mirror, and its image 1 unit right. In $y = - x$ : $(x, y) \mapsto (-y, -x)$ , so $Q^{'} = (- 1, - 5)$ .

Important

Rotation about any centre $X (a, b)$ . Rotate the vector from the centre, then add it back to the centre:

\overrightarrow{XP} = \begin{pmatrix} u \\ v \end{pmatrix} \quad\longmapsto\quad \overrightarrow{XP'} = \begin{pmatrix} -v \\ u \end{pmatrix} \ (+90^\circ), \quad \begin{pmatrix} -u \\ -v \end{pmatrix} \ (180^\circ), \quad \begin{pmatrix} v \\ -u \end{pmatrix} \ (-90^\circ).

Worked example 3: a quarter-turn about a point that is not the origin

Rotate triangle $A (3, 1)$ , $B (5, 1)$ , $C (5, 2)$ through $+90^\circ$ about $X (2, 1)$ .

Solution

Vertex	$\overrightarrow{XP}$	turned $+90^\circ$	Image
$A (3, 1)$	$\begin{pmatrix} 1 \\ 0 \end{pmatrix}$	$\begin{pmatrix} 0 \\ 1 \end{pmatrix}$	$A^{'} (2, 2)$
$B (5, 1)$	$\begin{pmatrix} 3 \\ 0 \end{pmatrix}$	$\begin{pmatrix} 0 \\ 3 \end{pmatrix}$	$B^{'} (2, 4)$
$C (5, 2)$	$\begin{pmatrix} 3 \\ 1 \end{pmatrix}$	$\begin{pmatrix} -1 \\ 3 \end{pmatrix}$	$C^{'} (1, 4)$

Check: $X B = X B^{'} = 3$ , and $\overrightarrow{XB} \cdot \overrightarrow{XB'} = 3 \times 0 + 0 \times 3 = 0$ , a right angle. ✓

Worked example 4: name the transformation

Describe fully the isometry with each rule: (a) $(x, y) \mapsto (y, -x)$ ; (b) $(x, y) \mapsto (4 - x,\; y)$ ; (c) $(x, y) \mapsto (2 - x,\; 6 - y)$ .

Solution (a) The rotation through $-90^\circ$ (a quarter-turn clockwise) about $O$ . (b) $4 - x = 2 \times 2 - x$ : the reflection in the line $x = 2$ . (c) $(2 \times 1 - x,\; 2 \times 3 - y)$ : the central symmetry about $(1, 3)$ .

Warning

A rotation needs three facts: centre, angle and direction. "Rotation through $90^\circ$ " alone is not a full description.

The rules $(- y, x)$ and $(y, - x)$ are for the origin only. For another centre, rotate the vector from the centre, as in Worked example 3.

Tip

To check a quarter-turn on squared paper, turn the page: after a $+90^\circ$ rotation about $X$ , each "3 right" from $X$ becomes "3 up", and each "1 up" becomes "1 left".

Mr Mathematics: Describing transformations. Watch for the facts each description needs: a vector for a translation, a mirror line for a reflection, and a centre, angle and direction for a rotation.

✦

Practice

The last two are harder.

Translate $A (4, - 1)$ by $\begin{pmatrix} -6 \\ 3 \end{pmatrix}$ .
Find the image of $(3, 5)$ under the central symmetry about $(2, 2)$ .
Reflect $(6, - 2)$ in (a) the line $y = x$ ; (b) the line $y = 1$ .
Rotate $(2, 5)$ about the origin through (a) $+90^\circ$ ; (b) $-90^\circ$ .
Rotate $P (4, 3)$ through $+90^\circ$ about $X (1, 2)$ .
Describe fully the isometry $(x, y) \mapsto (-x,\; 8 - y)$ .
A rotation about the origin maps $(5, 0)$ to $(0, - 5)$ and $(0, 2)$ to $(2, 0)$ . Describe it and find the image of $(- 3, 4)$ .

Show answersHide answers

$(- 2, 2)$ .
$(2 \times 2 - 3,\; 2 \times 2 - 5) = (1, -1)$ .
(a) $(- 2, 6)$ ; (b) $(6,\; 2 \times 1 - (-2)) = (6, 4)$ .
(a) $(- 5, 2)$ ; (b) $(5, - 2)$ .
$\overrightarrow{XP} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$ turns to $\begin{pmatrix} -1 \\ 3 \end{pmatrix}$ , so $P^{'} (0, 5)$ .
$(2 \times 0 - x,\; 2 \times 4 - y)$ : the central symmetry about $(0, 4)$ .
The rotation through $-90^\circ$ about $O$ , $(x, y) \mapsto (y, -x)$ ; $(-3, 4) \mapsto (4, 3)$ .

Sources and credits

Written by Ganzaa.
Photos: Imigongo traditional patterns (2) and (7), Ji-Elle, Wikimedia Commons, CC BY-SA 3.0.
Video: Describing Transformations, Mr Mathematics, embedded from YouTube.
Background on Imigongo: Imigongo, Wikipedia contributors, CC BY-SA 4.0, adapted by Ganzaa.

The diagrams, worked examples and practice items were written by Ganzaa. This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1Patterns on a wall

2The four isometries at a glance

Patterns on a wall

The four isometries at a glance