Geometric Transformations: Rotation, Reflection, Translation, and Enlargement

The Four Types of Transformation

There are four transformations you need to master. Let's start with a quick overview.

Now, let's dive into each transformation, focusing on both performing them and describing them using past-paper examples. For this, we'll rely heavily on video walkthroughs, as you prefer a practice-oriented approach.

1. Translation

A translation is the simplest transformation—it's simply sliding a shape from one position to another without changing its size or orientation.

Key Information: To describe a translation, you only need one piece of information: the translation vector.
The vector is written as $\left(\begin{array}{c}x\\ y\end{array}\right)$, where:

• $x$ is the movement left (negative) or right (positive).
• $y$ is the movement down (negative) or up (positive).

2. Reflection

A reflection "flips" a shape across a mirror line. Every point on the image is the same perpendicular distance from the mirror line as the corresponding point on the object.

Key Information: To describe a reflection, you need to state the equation of the mirror line.

Common mirror lines in exams include:

• The x-axis ($y=0$) or the y-axis ($x=0$).
• Horizontal lines ($y=k$) or vertical lines ($x=k$).
• Diagonal lines ($y=x$ or $y=-x$).

Let's see how to handle these.

Reflections in diagonal lines like $y=x$ can be trickier. A good strategy in the exam is to turn your paper so the mirror line is vertical or horizontal to you.

3. Rotation

A rotation "turns" a shape around a fixed point. This is often the trickiest transformation to describe fully.

Key Information: To describe a rotation, you need three details:

1. Angle of rotation: (e.g., 90°, 180°, 270°)
2. Direction of rotation: (Clockwise or Anticlockwise)
3. Centre of rotation: (A coordinate point (x, y))

Performing a rotation is easiest with tracing paper (which you can and should ask for in an exam), but describing one requires a specific method to find the centre.

4. Enlargement

An enlargement changes the size of a shape. It can get bigger or smaller, and can also be inverted.

Key Information: To describe an enlargement, you need two details:

1. Scale Factor: How many times larger or smaller the image is. It's calculated as $\text{Scale Factor}=\frac{\text{New Length}}{\text{Old Length}}$.
2. Centre of Enlargement: The fixed point from which the enlargement occurs.

Understanding Scale Factors:

• Positive SF > 1: The shape gets bigger.
• Positive SF between 0 and 1 (a fraction): The shape gets smaller.
• Negative SF: The shape is enlarged and appears on the opposite side of the centre, inverted (as if it has been rotated 180°).

Conclusion

You have now seen worked examples for performing and describing all four single transformations. Mastering this topic is about being methodical and precise. Always remember to provide all the required information to secure full marks.

Key Takeaways:

To fully describe a transformation, you must state:

• Translation: The translation vector, e.g., $\left(\begin{array}{c}-3\\ 5\end{array}\right)$.
• Reflection: The equation of the mirror line, e.g., "reflection in the line $y=x$".
• Rotation: The angle, direction, and centre of rotation, e.g., "rotation 90° anticlockwise about the point $(2,1)$".
• Enlargement: The scale factor and centre of enlargement, e.g., "enlargement with scale factor -2, centre $(0,0)$".

For tricky descriptions, remember the key strategies:

• Finding the Centre of Rotation: Use the perpendicular bisector method.
• Finding the Centre of Enlargement: Draw straight lines through corresponding vertices of the object and image.

Preview of the Next Lesson:
In this lesson, we saw that translations are described using vectors. In our next lesson, we will delve deeper into vector arithmetic. You'll learn how to add, subtract, and multiply vectors by scalars, and how to represent them using column notation. This will build the foundation for using vectors to solve more complex geometric problems.