What you'll learn
Throughout these notes a column vector is written in brackets as (x, y): the first number is the movement across and the second is the movement up or down.
Transformations and vectors form a substantial component of the CIE IGCSE Mathematics syllabus, tested across both Paper 2 and Paper 4. This topic combines geometric understanding with algebraic notation, requiring students to describe, perform, and combine transformations on 2D shapes, as well as manipulate column vectors and apply them to geometric problems. Mastery of this content typically accounts for 10-15 marks per examination paper.
Key terms and definitions
Transformation — a rule that moves, changes the size, or alters the orientation of a shape on a coordinate plane.
Object — the original shape before a transformation is applied.
Image — the resulting shape after a transformation has been applied, usually denoted with prime notation (A').
Invariant point — a point that remains in the same position after a transformation; its object and image coordinates are identical.
Vector — a quantity with both magnitude and direction, represented as a column vector (x, y) or using notation such as AB or a.
Resultant vector — the single vector obtained by adding two or more vectors together.
Position vector — a vector that describes the position of a point relative to the origin O, denoted OA for point A.
Scalar multiple — a vector multiplied by a scalar (number), which changes its magnitude and potentially its direction if the scalar is negative.
Core concepts
Translation
A translation moves every point of a shape the same distance in the same direction. Translations are described using column vectors.
The vector (a, b) means:
- Move a units horizontally (right if positive, left if negative)
- Move b units vertically (up if positive, down if negative)
To fully describe a translation on an exam:
- State "translation"
- Give the column vector (a, b)
Translations have no invariant points (unless the vector is (0, 0)). The object and image are congruent and have the same orientation.
Reflection
A reflection creates a mirror image of a shape across a line called the mirror line or axis of reflection. Each point on the image is the same perpendicular distance from the mirror line as the corresponding point on the object.
Common mirror lines tested in CIE IGCSE Mathematics:
- The x-axis (equation y = 0)
- The y-axis (equation x = 0)
- The line y = x
- The line y = -x
- Horizontal lines: y = k (where k is a constant)
- Vertical lines: x = k
To fully describe a reflection:
- State "reflection"
- Give the equation of the mirror line
Points on the mirror line are invariant. The object and image are congruent but opposite in orientation (one is a mirror image of the other).
Rotation
A rotation turns a shape about a fixed point called the centre of rotation through a specified angle.
To fully describe a rotation, three pieces of information are required:
- State "rotation"
- Give the centre of rotation as coordinates (a, b)
- State the angle of rotation (with direction: clockwise or anticlockwise)
Common angles: 90°, 180°, 270°. The direction matters for angles other than 180°.
The centre of rotation is the only invariant point (unless the angle is 360° or 0°). To find the centre of rotation:
- Join corresponding points on object and image
- Draw perpendicular bisectors of these line segments
- The centre is where the perpendicular bisectors intersect
The object and image are congruent.
Enlargement
An enlargement changes the size of a shape using a centre of enlargement and a scale factor.
To fully describe an enlargement:
- State "enlargement"
- Give the centre of enlargement as coordinates (a, b)
- State the scale factor k
Scale factor effects:
- k > 1: image is larger than object
- 0 < k < 1: image is smaller than object (sometimes called a reduction)
- k = 1: image is identical to object
- k < 0: image is on the opposite side of the centre of enlargement and inverted
If scale factor is k, all lengths are multiplied by |k| and the area is multiplied by k².
To enlarge a shape:
- Draw lines from the centre of enlargement through each vertex of the object
- Multiply the distance from the centre to each vertex by the scale factor
- Mark the image vertices at these new distances
The centre of enlargement is the only invariant point. The object and image are similar shapes.
Combined transformations
CIE IGCSE Mathematics frequently tests the ability to perform one transformation followed by another. The order matters: transformation A followed by B usually produces a different result than B followed by A.
Notation: Transformation B followed by transformation A can be written as AB (read right to left, like function notation).
A single transformation that has the same effect as two successive transformations is called the equivalent single transformation. Examiners often ask students to describe this.
Column vectors
Vectors are represented as (x, y) where x is the horizontal component and y is the vertical component.
Vector addition: Add corresponding components
(a, b) + (c, d) = (a+c, b+d)
Vector subtraction: Subtract corresponding components
(a, b) − (c, d) = (a-c, b-d)
Scalar multiplication: Multiply each component by the scalar
k(a, b) = (ka, kb)
Magnitude (length) of a vector: For vector (a, b), the magnitude is:
|v| = √(a² + b²)
This uses Pythagoras' theorem.
Vector geometry
Vectors are used to solve geometric problems involving position and displacement.
Finding a vector between two points: If A has position vector a and B has position vector b, then:
AB = b − a
Parallel vectors: Vectors are parallel if one is a scalar multiple of the other. If u = kv where k is a scalar, then u and v are parallel.
Midpoint using vectors: The position vector of the midpoint M of AB is:
m = (a + b)/2
Ratio problems: If point P divides AB in the ratio m:n, then:
OP = (na + mb)/(m+n)
CIE IGCSE Mathematics questions often involve using vector methods to prove geometric properties such as points being collinear (on the same straight line) or shapes being parallelograms.
Worked examples
Example 1: Triangle ABC has vertices A(2, 1), B(4, 1), C(4, 3).
(a) Reflect triangle ABC in the line y = -x to give triangle A'B'C'. Write down the coordinates of A', B', and C'. [3 marks]
(b) Describe fully the single transformation that maps triangle ABC onto triangle A''B''C'' where A''(-1, -2), B''(-1, -4), C''(-3, -4). [3 marks]
Solution:
(a) For reflection in y = -x, the rule is (x, y) → (-y, -x):
- A(2, 1) → A'(-1, -2) ✓
- B(4, 1) → B'(-1, -4) ✓
- C(4, 3) → C'(-3, -4) ✓
(b) Comparing ABC to A''B''C'':
- A(2, 1) → A''(-1, -2): this is the same as A' from part (a)
- The transformation is a reflection ✓
- in the line y = -x ✓✓ (equation needed for full marks)
Example 2: The position vectors of points P and Q are p = (3, -2) and q = (7, 4).
(a) Find PQ. [2 marks]
(b) Find the magnitude of PQ. [2 marks]
(c) Point M is the midpoint of PQ. Find the position vector of M. [2 marks]
Solution:
(a) PQ = q − p = (7, 4) − (3, -2) ✓
= (4, 6) ✓
(b) |PQ| = √(4² + 6²) = √(16 + 36) = √52 ✓
= 7.21 (3 s.f.) or 2√13 ✓
(c) m = (p + q)/2 = ½(10, 2) ✓
= (5, 1) ✓
Example 3: Shape P is transformed by an enlargement, centre (1, 2), scale factor -2, to give shape Q. Shape Q is then reflected in the x-axis to give shape R. Describe fully the single transformation that maps shape P directly onto shape R. [3 marks]
Solution:
This requires working through both transformations:
- The enlargement with scale factor -2 enlarges and inverts the shape
- The reflection in the x-axis flips the shape
- Combined effect: enlargement ✓, centre (1, 2) ✓, scale factor 2 ✓
(The negative scale factor and reflection cancel each other's inversion effects)
Common mistakes and how to avoid them
Mistake: Describing a transformation incompletely, for example stating "rotation, 90°" without the centre or direction. Correction: Always include all required information: reflection needs mirror line equation; rotation needs centre, angle, and direction; enlargement needs centre and scale factor; translation needs the complete column vector.
Mistake: Confusing AB with BA, or calculating a − b when b − a is needed. Correction: Remember AB means "from A to B", calculated as (position of B) minus (position of A). The order matters: BA = -AB.
Mistake: When reflecting in y = x or y = -x, simply swapping coordinates without considering signs. Correction: For y = x: (x, y) → (y, x). For y = -x: (x, y) → (-y, -x). Plot one point carefully to check.
Mistake: Stating clockwise rotation when the rotation is anticlockwise, or vice versa. Correction: Trace the rotation with your finger or pencil from object to image. By convention, anticlockwise is positive. A 90° clockwise rotation is equivalent to 270° anticlockwise.
Mistake: Calculating area scale factor as k instead of k² for an enlargement with scale factor k. Correction: Linear dimensions multiply by k; areas multiply by k²; volumes multiply by k³. If scale factor is 3, the area becomes 3² = 9 times larger.
Mistake: Adding vectors incorrectly when solving geometry problems, particularly in ratio questions or when dealing with multiple vectors. Correction: Draw a clear diagram with all vectors marked. Break the problem into steps. Use the triangle law: to go from A to C via B, AC = AB + BC.
Exam technique for Transformations and Vectors
Command word "Describe fully": This requires all necessary information for the transformation type. Missing any component (such as omitting the direction for a rotation or giving only the scale factor without the centre for an enlargement) will lose marks. Write the transformation name first, then give each required detail on the same line or clearly below.
Drawing transformations: Use a sharp pencil and ruler for accuracy. When enlarging, draw construction lines from the centre through each vertex. For reflections, count squares carefully to ensure equal perpendicular distances. Label the image clearly with the requested letter (usually with a prime, such as A'B'C'). Accuracy is typically within 1mm on CIE mark schemes.
Vector working: Show all working when performing vector arithmetic. Write column vectors neatly in brackets, keeping the across-number first and the up-or-down number second. When finding magnitudes, always show the substitution into the formula √(x² + y²) before calculating. Examiners award method marks even if the final answer is incorrect.
Combined transformations: When describing an equivalent single transformation, perform each transformation carefully on graph paper, then describe what single transformation would achieve the same result. Check all three components (where applicable) match the image position exactly.
Quick revision summary
Four transformations exist: translation (column vector needed), reflection (equation of mirror line needed), rotation (centre, angle, direction needed), and enlargement (centre and scale factor needed). Vectors combine by adding/subtracting components; magnitude uses Pythagoras. Vector AB = b − a. Parallel vectors are scalar multiples. Combined transformations may simplify to a single equivalent transformation. Always describe transformations fully with all required information for full marks.