Geometrical Transformations

Core Concepts

Geometrical transformations involve moving a shape or a point in the Cartesian plane according to specific mathematical rules. A transformation maps an object (the original shape or point) to an image (the new shape or point). The main types of transformations covered in this syllabus are reflection, rotation, translation, and enlargement.

1. Reflection

A reflection is a transformation that "flips" a figure over a specific line, known as the mirror line or line of reflection. Each point on the object and its corresponding point on the image are equidistant from the mirror line, and the line connecting them is perpendicular to the mirror line.

Common reflections in the Cartesian plane include:

Reflection in the $x$-axis (the line $y = 0$): A point $(x, y)$ is mapped to $(x, -y)$.
Reflection in the $y$-axis (the line $x = 0$): A point $(x, y)$ is mapped to $(-x, y)$.
Reflection in the line $y = x$: A point $(x, y)$ is mapped to $(y, x)$.
Reflection in the line $y = -x$: A point $(x, y)$ is mapped to $(-y, -x)$.

2. Rotation

A rotation turns a figure about a fixed point called the center of rotation, through a specific angle, called the angle of rotation. The direction of rotation is either clockwise or anticlockwise. By convention, an anticlockwise rotation is considered positive, and a clockwise rotation is negative.

Common rotations about the origin $(0, 0)$ include:

$90^{\circ}$ Anticlockwise (or $270^{\circ}$ Clockwise): A point $(x, y)$ maps to $(-y, x)$.
$180^{\circ}$ (Half-turn): A point $(x, y)$ maps to $(-x, -y)$.
$270^{\circ}$ Anticlockwise (or $90^{\circ}$ Clockwise): A point $(x, y)$ maps to $(y, -x)$.
General Angle $\theta$ Anticlockwise: A point $(x, y)$ maps to $(x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)$.

3. Translation

A translation "slides" an object by a specific distance in a given direction without changing its size or orientation. A translation can be described by a translation vector $T = \begin{pmatrix} a \\ b \end{pmatrix}$, where $a$ represents the horizontal shift and $b$ represents the vertical shift.

If a point $P(x, y)$ undergoes a translation by vector $T$, its image $P'(x', y')$ is given by: $$ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} x + a \\ y + b \end{pmatrix} $$

4. Enlargement

Enlargement (or dilation) changes the size of a figure but not its shape. It requires a center of enlargement and a scale factor, $k$. The scale factor determines how much larger or smaller the image will be compared to the object.

If $|k| > 1$, the image is larger than the object.
If $0 < |k| < 1$, the image is smaller.
If $k$ is negative, the image is inverted and appears on the opposite side of the center of enlargement.

For an enlargement from the origin $(0,0)$ with scale factor $k$, a point $(x, y)$ maps to $(kx, ky)$. If the center of enlargement is $(a, b)$, the transformation is given by: $$ \begin{pmatrix} x' \\ y' \end{pmatrix} = k \begin{pmatrix} x - a \\ y - b \end{pmatrix} + \begin{pmatrix} a \\ b \end{pmatrix} $$

5. Combined Transformations

A combined (or composite) transformation involves applying two or more transformations in a sequence. The order in which transformations are applied is crucial because matrix operations for these transformations are generally not commutative. If transformation $A$ is followed by transformation $B$, the combined operation is evaluated as applying $B$ to the result of $A$.

Worked Examples

Example 1: Translation

Question: A translation takes point $(5, 5)$ to the point $(-7, -7)$. If it takes point $(x, y)$ to $(-4, -4)$, find the values of $x$ and $y$.

Solution: Let the translation vector be $T = \begin{pmatrix} a \\ b \end{pmatrix}$. Using the first point and its image: $$ \begin{pmatrix} 5 \\ 5 \end{pmatrix} + \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} -7 \\ -7 \end{pmatrix} $$ Solving for $a$ and $b$: $$ a = -7 - 5 = -12 $$ $$ b = -7 - 5 = -12 $$ So, the translation vector is $T = \begin{pmatrix} -12 \\ -12 \end{pmatrix}$.

Now, apply this translation vector to find the object $(x, y)$ whose image is $(-4, -4)$: $$ \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} -12 \\ -12 \end{pmatrix} = \begin{pmatrix} -4 \\ -4 \end{pmatrix} $$ $$ x - 12 = -4 \implies x = 8 $$ $$ y - 12 = -4 \implies y = 8 $$ Therefore, the values are $x = 8$ and $y = 8$.

Example 2: General Rotation

Question: Find the image of the point $A(4, 2)$ after a rotation about the origin through $120^\circ$ anticlockwise.

Solution: The general formula for an anticlockwise rotation by an angle $\theta$ about the origin is: $$ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} $$ Given $\theta = 120^\circ$, we have $\cos 120^\circ = -0.5$ and $\sin 120^\circ = \frac{\sqrt{3}}{2}$. The coordinates of $A$ are $(4, 2)$. $$ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} -0.5 & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -0.5 \end{pmatrix} \begin{pmatrix} 4 \\ 2 \end{pmatrix} $$ Calculate the new coordinates: $$ x' = -0.5(4) - \frac{\sqrt{3}}{2}(2) = -2 - \sqrt{3} $$ $$ y' = \frac{\sqrt{3}}{2}(4) - 0.5(2) = 2\sqrt{3} - 1 $$ Therefore, the image of $A$ is $A'(-2 - \sqrt{3}, 2\sqrt{3} - 1)$.

Example 3: Combined Transformations

Question: Point $A(4, 2)$ is reflected in the line $y + x = 0$ followed by an anticlockwise rotation through $90^\circ$ about the origin. Find the final image of point $A$.

Solution: Step 1: Reflection in the line $y + x = 0$ (which is $y = -x$) The rule for reflection in the line $y = -x$ is that point $(x, y)$ maps to $(-y, -x)$. Applying this to $A(4, 2)$: $$ A_1 = (-2, -4) $$

Step 2: Anticlockwise rotation through $90^\circ$ about the origin The rule for a $90^\circ$ anticlockwise rotation is that point $(x, y)$ maps to $(-y, x)$. Applying this rule to our intermediate point $A_1(-2, -4)$: $$ A_2 = (-(-4), -2) = (4, -2) $$ The final image of point $A$ is $(4, -2)$.

NECTA Exam Focus

Based on recent NECTA past papers, questions on Geometrical Transformations frequently assess a student's ability to track the coordinates of points across single or multiple transformation steps.

Recurring Themes:

Coordinate Geometry Integration: Questions are almost exclusively set on the Cartesian plane, requiring algebraic manipulation of vectors or coordinates rather than purely geometric construction.
Combined Transformations: NECTA strongly favors composite transformations. You will commonly see a reflection followed by a rotation, or multiple sequential rotations.
Translation Vectors: Students are often asked to determine an unknown translation vector from an object-image pair, and then apply it to another coordinate.
Standard Angles of Rotation: Rotations typically involve standard angles ($90^\circ, 180^\circ, 270^\circ$). However, angles requiring basic trigonometry (like $120^\circ$) also appear, necessitating knowledge of exact trigonometric ratios.

Common Pitfalls:

Order of Operations: In combined transformations, calculating them in reverse order is a frequent mistake. Transformation $A$ followed by $B$ means evaluating the coordinates after $A$, and applying $B$ to that result.
Sign Errors: Mistakes with negative signs are common, particularly when reflecting across the line $y = -x$ or applying the algebraic rules for rotations.
Clockwise vs. Anticlockwise: Mixing up the rotation directions. Remember that anticlockwise is generally considered mathematically positive, though the specific rules map coordinates differently.
Trigonometric Values: For non-right-angle rotations (e.g., $120^\circ$), students sometimes substitute incorrect values for sine and cosine.

Practice Problems

A translation takes point $(5, 5)$ to the point $(-7, -7)$. If it takes point $(x, y)$ to $(-4, -4)$, find the values of $x$ and $y$.
Point $A(4, 2)$ is reflected in the line $y + x = 0$ followed by an anticlockwise rotation through $90^\circ$ about the origin. Find the final image of point $A$.
Find the image of the point A(4, 2) after a rotation about the origin through $120^\circ$ anticlockwise.
An initial point $(7, 4)$ undergoes a rotation of $90^{\circ}$ followed by another rotation of $180^{\circ}$ anticlockwise. Determine its final image coordinates.

Subtopics

Reflection
Rotation
Translation
Enlargement
Combined transformations

Crosswalk Notes

Cross-version relationships are drafted in data/curricula/crosswalks/csee-basic-mathematics-2005-to-mathematics-2023.json. Partial and 2005-only mappings remain reviewable.