International Mathematics (0607) | Transformations

Hello IGCSE Math Student! Welcome to Transformations (0607)

Welcome to Chapter 8: Transformations! This is a fascinating topic where we learn how to move shapes around the coordinate plane without changing their fundamental characteristics.

Geometry often feels like a series of fixed rules, but transformations show us the dynamic side of shapes—how they can be reflected, rotated, enlarged, and translated. Mastering these concepts is essential for understanding spatial relationships and forms a solid foundation for more advanced geometry and vectors later on.

Don't worry if visualizing these moves seems tricky at first. We will use simple rules and coordinates to break down every type of movement!

Section 1: The Four Types of Transformation

In IGCSE Mathematics, there are four key types of transformation you must be able to recognize, describe, and draw: Translation, Reflection, Rotation, and Enlargement.

1. Translation (The Slide)

A translation simply means sliding a shape from one position to another without turning or flipping it.

• A translation is fully described by a single piece of information: the Translation Vector.
• The vector is written as a column matrix: \( \mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix} \).

What the Vector Means:
\( x \): Movement in the horizontal direction. (Positive = Right, Negative = Left)
\( y \): Movement in the vertical direction. (Positive = Up, Negative = Down)

Example: If you translate a point P(2, 5) by the vector \( \begin{pmatrix} 3 \\ -1 \end{pmatrix} \), the new point P' is \((2+3, 5-1) = (5, 4)\).

Quick Review: Translation

To Describe a Translation: You only need to state the translation vector \(\begin{pmatrix} x \\ y \end{pmatrix}\).

2. Reflection (The Flip)

Reflection is flipping a shape over a mirror line. The distance from the object to the mirror line is exactly the same as the distance from the mirror line to the image.

• Every point in the image is equidistant from the mirror line as the corresponding point in the object.
• The line connecting a point and its image is perpendicular to the mirror line.

Key Requirement (Core C8.1): You must be able to reflect shapes in vertical lines (e.g., \(x = 2\)) and horizontal lines (e.g., \(y = -3\)).

Step-by-Step Reflection:

1. Identify the mirror line (M).
2. Choose a vertex (P) on the object.
3. Measure the perpendicular distance from P to M.
4. Move the same distance on the opposite side of M along the same perpendicular line to find the image point (P').
5. Repeat for all vertices and join them up.

Accessibility Tip: Counting Squares

When reflecting in lines like \(x=k\) or \(y=k\), you can simply count the perpendicular distance (squares) from the point to the line, and count the same number of squares past the line.

Extended Note (E8.1): Extended students must be able to reflect in any straight line, such as \(y = x\), \(y = -x\), or \(y = 2x + 1\). If the mirror line is diagonal, remember that the distance must be measured perpendicular to the mirror line. Using tracing paper can be very helpful here!

Section 2: Rotation (The Turn)

Rotation is turning a shape around a fixed point called the Centre of Rotation.

To fully describe a rotation, you need three pieces of information:

1. Centre of Rotation (a coordinate, e.g., \((0, 0)\) or \((1, 4)\)).
2. Angle of Rotation (e.g., \(90^{\circ}\), \(180^{\circ}\), \(270^{\circ}\)).
3. Direction (e.g., clockwise (CW) or anticlockwise (ACW)).

Syllabus Focus (C8.1 & E8.1): You only need to deal with rotations through multiples of \(90^{\circ}\) (which means \(90^{\circ}\), \(180^{\circ}\), \(270^{\circ}\)).

Key Rotation Facts

• A \(180^{\circ}\) rotation is the same whether clockwise or anticlockwise.
• A \(90^{\circ}\) clockwise rotation is the same as a \(270^{\circ}\) anticlockwise rotation.

Step-by-Step Rotation (Using Tracing Paper or Ruler/Protractor)

(A ruler must be used for all straight edges in the exam.)

1. Mark the Centre (C) of Rotation clearly.
2. Connect a Vertex (P) of the object to the Centre (C).
3. Rotate the Line Segment PC by the required angle and direction around C. (Use a protractor for accuracy, or count squares for \(90^{\circ}\) and \(180^{\circ}\) rotations around simple centres like the origin).
4. Mark the Image Point (P'), ensuring the distance CP' is equal to CP.
5. Repeat this process for all vertices.

Memory Aid for 90° Rotation about the Origin \((0, 0)\)

If point P is \((x, y)\):

• \(90^{\circ}\) Anticlockwise: \((x, y) \rightarrow (-y, x)\)
• \(90^{\circ}\) Clockwise: \((x, y) \rightarrow (y, -x)\)
• \(180^{\circ}\) (Both directions): \((x, y) \rightarrow (-x, -y)\)

Section 3: Enlargement (The Stretch)

Enlargement changes the size of a shape but keeps its angles and shape exactly the same (the object and image are similar).

To fully describe an enlargement, you need two pieces of information:

1. Centre of Enlargement (C) (a coordinate).
2. Scale Factor (SF) (a number).

Scale Factor Rules

• If SF > 1, the image is bigger than the object.
• If $0 < SF < 1$, the image is smaller than the object (a reduction).
• If SF is positive, the image is on the same side of the centre as the object.

Core Syllabus (C8.1) Limitation: Core students only deal with Positive and Fractional Scale Factors (e.g., 2, 0.5, 1/4).

Extended Syllabus (E8.1) Requirement: Extended students must also handle Negative Scale Factors.

What Negative Scale Factors Mean

If the scale factor (SF) is negative, the enlargement is inverted and appears on the opposite side of the Centre of Enlargement (C).

Example: An enlargement with SF = -2 means the image is twice as large, but upside down and reflected through the centre.

Step-by-Step Enlargement using Coordinates

If $C$ is the Centre of Enlargement \((x_c, y_c)\) and $P$ is a point \((x_p, y_p)\) on the object, the image point $P'$ is calculated using the formula:

\( \vec{CP'} = SF \times \vec{CP} \)

In words, you find the vector from the centre to the point, multiply the vector by the SF, and then add this new vector back to the centre coordinates.

1. Identify C and SF.
2. Choose a vertex P. Find the movement from C to P (horizontal change, vertical change).
3. Multiply these movements by the SF.
4. Apply the new multiplied movements starting from C to find P'.
5. Repeat for all vertices.

Did You Know?

The ratio of the Area of the image to the Area of the object is equal to \( (\text{Scale Factor})^2 \).

Section 4: Extended Content - Vectors in Two Dimensions (E8.2 & E8.3)

Vectors are essential tools for describing movement (Translation) and forces, as they have both magnitude (size) and direction.

1. Vector Notation

A vector can be represented in three ways:

• Column Vector: \( \mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix} \). (Used primarily for translation).
• Directed Line Segment: \( \vec{AB} \). (Starts at A, ends at B).
• Bold/Underlined Letter: \(\mathbf{a}\) (used in printed text).

2. Vector Operations

Vector Addition and Subtraction

When adding or subtracting vectors, you simply add or subtract the corresponding components (x components with x components, y components with y components).

If \( \mathbf{a} = \begin{pmatrix} 2 \\ 5 \end{pmatrix} \) and \( \mathbf{b} = \begin{pmatrix} -1 \\ 3 \end{pmatrix} \):
\( \mathbf{a} + \mathbf{b} = \begin{pmatrix} 2 + (-1) \\ 5 + 3 \end{pmatrix} = \begin{pmatrix} 1 \\ 8 \end{pmatrix} \)
\( \mathbf{a} - \mathbf{b} = \begin{pmatrix} 2 - (-1) \\ 5 - 3 \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} \)

Analogy: Vector Journeys

Think of vectors as a journey. To find \( \mathbf{a} + \mathbf{b} \), you take the journey \(\mathbf{a}\) followed by the journey \(\mathbf{b}\). Vector addition follows the "head-to-tail" rule on a diagram.

Scalar Multiplication

Multiplying a vector by a scalar (a simple number, like 2 or -3) changes its magnitude, and sometimes its direction.

If \( \mathbf{a} = \begin{pmatrix} 4 \\ -2 \end{pmatrix} \) and we multiply by scalar 3:
\( 3\mathbf{a} = 3 \times \begin{pmatrix} 4 \\ -2 \end{pmatrix} = \begin{pmatrix} 3 \times 4 \\ 3 \times (-2) \end{pmatrix} = \begin{pmatrix} 12 \\ -6 \end{pmatrix} \)

Note: If the scalar is negative (e.g., \(-2\)), the resulting vector points in the opposite direction.

3. Magnitude of a Vector (E8.3)

The magnitude of a vector is its length. Since a vector \(\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}\) forms a right-angled triangle on the coordinate grid (with sides \(x\) and \(y\)), we use Pythagoras' theorem to find its length.

The magnitude is denoted by vertical lines, e.g., \( |\mathbf{v}| \) or \( |\vec{AB}| \).

Magnitude Formula: \( |\mathbf{v}| = \sqrt{x^2 + y^2} \)

Example: Find the magnitude of \( \mathbf{v} = \begin{pmatrix} 3 \\ -4 \end{pmatrix} \).
\( |\mathbf{v}| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)

Section 5: Advanced Transformations (Extended E8.1 Notes)

1. Describing the Reverse Transformation

The reverse (or inverse) of a transformation takes the image back exactly to the object's position.

• Reverse of Translation: Use the negative of the vector.
  E.g., Reverse of Translation by \( \begin{pmatrix} 4 \\ -2 \end{pmatrix} \) is Translation by \( \begin{pmatrix} -4 \\ 2 \end{pmatrix} \).
• Reverse of Reflection: A reflection is its own reverse. The mirror line stays the same.
• Reverse of Rotation: Keep the same centre, keep the same angle, but reverse the direction.
  E.g., Reverse of \(90^{\circ}\) ACW is \(90^{\circ}\) CW.
• Reverse of Enlargement: Keep the same centre, but use the reciprocal of the scale factor.
  E.g., Reverse of Enlargement SF=3 is Enlargement SF=1/3. Reverse of SF=-2 is SF=-1/2.

2. Combinations of Transformations (Extended Only)

The Core syllabus specifically states that questions will not involve combinations. However, Extended students must be prepared for two or more transformations applied consecutively.

If a shape \(A\) is transformed onto \(B\), and then \(B\) is transformed onto \(C\), you need to follow the steps in order: \( A \rightarrow B \rightarrow C \).

Sometimes you will be asked to find the single transformation that maps \(A\) onto \(C\).
Example: A reflection followed by another reflection might result in a single translation.