Hello IGCSE Maths Students! Welcome to Transformations!

Geometry can be fun, especially when you get to move things around! This chapter is all about changing the position or size of a shape on a coordinate grid. Think of it like choreography—you’re telling a shape exactly how to move.

Mastering transformations (moving shapes) is vital for coordinate geometry and preparing you for the world of vectors later on. Don't worry if it looks complicated; each move follows strict, simple rules. We'll break them down step-by-step!

1. Translation (The Simple Slide)

Translation is the simplest transformation—it just means sliding a shape from one position to another without changing its size, shape, or orientation (which way it is facing).

What you need to describe a Translation:

You only need one piece of information: the Translation Vector.

A translation vector is written like this: \(\begin{pmatrix} x \\ y \end{pmatrix}\).

  • The top number (x) tells you the movement horizontally (left or right).
  • The bottom number (y) tells you the movement vertically (up or down).
Rules for the Vector \(\begin{pmatrix} x \\ y \end{pmatrix}\):
  • If \(x\) is positive, move Right.
  • If \(x\) is negative, move Left.
  • If \(y\) is positive, move Up.
  • If \(y\) is negative, move Down.

Example: A shape is translated by the vector \(\begin{pmatrix} 3 \\ -2 \end{pmatrix}\).

This means: Move every point on the shape 3 units Right and 2 units Down.

Key Takeaway:

Translation is defined entirely by a vector, telling you exactly how far and in which direction to slide the shape. The shape itself remains congruent (identical size and shape).

2. Reflection (The Mirror Image)

Reflection is the transformation that produces a mirror image of a shape across a fixed line.

What you need to describe a Reflection:

You must specify the Line of Reflection (or Mirror Line).

How to Perform a Reflection:

Imagine folding the paper along the mirror line. The key rule is that:

  1. Every point on the image must be the same perpendicular distance from the mirror line as the corresponding point on the object.
Syllabus Scope: Core vs. Extended
For CORE (C8.1):

You only need to deal with reflections in vertical lines or horizontal lines.

  • Vertical Lines: Always have the equation \(x = k\) (e.g., \(x=1\), \(x=-3\)).
  • Horizontal Lines: Always have the equation \(y = k\) (e.g., \(y=2\), \(y=0\) which is the x-axis).
For EXTENDED (E8.1):

You may need to reflect in any straight line, such as slanted lines like \(y=x\).

  • Tip for \(y=x\): Swap the coordinates. If Object P is (2, 5), Image P' is (5, 2).

Common Mistake to Avoid:

Students often forget that the line of reflection does not have to be an axis! A reflection in the line \(x=3\) is a vertical line passing through \(x=3\), not the y-axis.

Key Takeaway:

Reflection requires defining the mirror line. The image is congruent and flipped (the orientation is reversed).

3. Rotation (The Turn)

Rotation means turning a shape around a fixed point.

What you need to describe a Rotation:

You must specify three things:

  1. The Centre of Rotation (the fixed point the shape turns around).
  2. The Angle of Rotation (how far it turns).
  3. The Direction of Rotation (Clockwise or Anti-clockwise).
Direction Rules:
  • Anti-Clockwise (ACW): Standard positive direction (e.g., \(+90^\circ\)).
  • Clockwise (CW): Negative direction (e.g., \(-90^\circ\)).
  • Note: A \(90^\circ\) CW rotation is the same as a \(270^\circ\) ACW rotation. A \(180^\circ\) rotation has no direction, as it looks the same either way.
Syllabus Scope: Core vs. Extended
For CORE (C8.1):

You only need rotations that are multiples of 90° (i.e., 90°, 180°, or 270°). The centre is often the origin (0, 0), a vertex, or the midpoint of an edge.

For EXTENDED (E8.1):

The angle can be any degree (e.g., \(30^\circ\)) and the centre can be any point.

Memory Aid: Using Tracing Paper

This is the most reliable method for both performing and describing rotations:

  1. Place tracing paper over the grid and trace the object shape.
  2. Mark the Centre of Rotation clearly with your pencil.
  3. Place the point of your pencil firmly on the centre of rotation.
  4. Rotate the tracing paper by the required angle and direction.
  5. Mark the position of the new image (A', B', C', etc.) onto the original grid.

How to FIND the Centre of Rotation (To Describe the Transformation):

If you are given the object and the image, you can find the centre by:

  1. Draw a straight line connecting a point on the object (A) to its corresponding point on the image (A').
  2. Find the perpendicular bisector of this line (a line perpendicular to AA' that cuts it in half).
  3. Repeat this for a second pair of points (B and B').
  4. The point where the two perpendicular bisectors cross is the Centre of Rotation.

Key Takeaway:

Rotation is defined by three factors: centre, angle, and direction. The resulting image is congruent to the object.

4. Enlargement (Changing Size)

Enlargement changes the size of a shape. It can make it bigger or smaller, but critically, it keeps the shape mathematically similar (all angles remain the same).

What you need to describe an Enlargement:

You must specify two things:

  1. The Centre of Enlargement.
  2. The Scale Factor (k).

The Scale Factor (k)

The scale factor determines how much larger or smaller the shape becomes. It is calculated as:

$$k = \frac{\text{Image Length}}{\text{Object Length}}$$

Syllabus Scope: Core vs. Extended
For CORE (C8.1):

The scale factor \(k\) will be positive and may be fractional (e.g., \(k=2\) or \(k=1/2\)).

  • If \(k > 1\), the shape gets larger.
  • If \(0 < k < 1\), the shape gets smaller (this is sometimes called a reduction).
For EXTENDED (E8.1):

The scale factor \(k\) can be negative (e.g., \(k=-2\)).

  • If \(k\) is negative, the image appears on the opposite side of the centre of enlargement and is rotated by 180°.
  • Example: Enlargement by scale factor \(-2\) from (0, 0). If a point is 3 units right of the centre, the image will be \(3 \times 2 = 6\) units left of the centre.

Step-by-Step: Performing an Enlargement

  1. Identify the Centre of Enlargement (C).
  2. Draw a straight line from C through one vertex of the object (A).
  3. Count the distance from C to A (the vector). For example, C to A is \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\).
  4. Multiply this vector by the scale factor \(k\). If \(k=3\), the new vector is \(3 \times \begin{pmatrix} 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 6 \\ 3 \end{pmatrix}\).
  5. Plot the new point A' starting from C using the calculated new vector.
  6. Repeat for all vertices.

Area and Volume Effects (Extended Focus)

If the shape is enlarged by a linear scale factor \(k\):

  • The Area of the image will be \(k^2\) times the Area of the object.
  • The Volume of the image (if 3D) will be \(k^3\) times the Volume of the object.

Did you know? If you enlarge a photo on your phone by a factor of 2, the area of the photo is actually 4 times bigger!

Key Takeaway:

Enlargement is defined by the centre and scale factor \(k\). Remember that a negative scale factor flips the shape through the centre.

5. Extended Topics and Reverse Transformations (E8.1)

For Extended candidates, you must be ready to deal with more complex scenarios.

A. Reverse Transformations

Sometimes you are asked to find the transformation that maps the Image back onto the Object. This is the reverse transformation.

  • Reverse Translation: Use the negative of the original vector.

    If Object \(\to\) Image uses \(\begin{pmatrix} 4 \\ -1 \end{pmatrix}\), then Image \(\to\) Object uses \(\begin{pmatrix} -4 \\ 1 \end{pmatrix}\).

  • Reverse Reflection: It's the same reflection line. (The mirror works both ways!)
  • Reverse Rotation: Keep the same centre, but use the opposite angle/direction.

    If Object \(\to\) Image is \(90^\circ\) ACW, then Image \(\to\) Object is \(90^\circ\) CW (or \(270^\circ\) ACW).

  • Reverse Enlargement: Keep the same centre, but use the reciprocal of the scale factor.

    If Object \(\to\) Image uses \(k=3\), then Image \(\to\) Object uses \(k=1/3\). If the original \(k=-2\), the reverse is \(k=-1/2\).

B. Combinations of Transformations (Describing a Single Result)

The Core syllabus specifically excludes asking about combinations. However, Extended candidates should be aware that two simple transformations might combine to look like a single, different transformation.

  • A translation followed by another translation is just one larger translation.
  • Two reflections over parallel lines result in a translation.
  • Two reflections over intersecting lines result in a rotation.

Don't worry about calculating complex combinations, but be prepared if the question asks you to "Describe the single transformation that maps shape A onto shape B," and that transformation is not obvious (e.g., finding the centre and scale factor of a complex enlargement).

Quick Review: The Four Transformations

Transformation Checklist (What to include when DESCRIBING):

  • Translation: Vector \(\begin{pmatrix} x \\ y \end{pmatrix}\)
  • Reflection: Line of Reflection (equation, e.g., \(y=3\))
  • Rotation: Centre (coordinate), Angle, Direction (CW/ACW)
  • Enlargement: Centre (coordinate), Scale Factor (k)

Remember that when you draw transformations in the exam, you must use a ruler for all straight edges. Good luck—you've got this!