Transformations and vectors for IGCSE Mathematics: Methods and Worked Examples

Two tools, one coordinate plane

Transformations and vectors occupy the same coordinate plane, but they ask different questions. Transformations ask: how does a shape move, flip, turn, or resize? Vectors ask: how do we describe displacement and prove geometric relationships algebraically? Both appear regularly on IGCSE Mathematics papers, and both reward precision. A vague description of a transformation loses marks just as surely as an incorrect vector expression.

This guide covers the four standard transformations (translation, reflection, rotation, enlargement), then moves into vector geometry. Core candidates need the transformations. Extended candidates need both.

Translation

A translation slides every point of a shape by the same distance in the same direction. It is described by a column vector.

The column vector (3, -2) means "3 units right, 2 units down." The top number is horizontal movement (positive = right, negative = left). The bottom number is vertical movement (positive = up, negative = down).

Worked example 1

Triangle P has vertices at (1, 2), (4, 2), and (4, 5). It is translated by the vector (−3, 1). Find the vertices of the image.

1. Add the vector to each vertex separately.
2. (1, 2) + (−3, 1) = (−2, 3)
3. (4, 2) + (−3, 1) = (1, 3)
4. (4, 5) + (−3, 1) = (1, 6)

The image has vertices (−2, 3), (1, 3), and (1, 6).

Reflection

A reflection flips a shape across a mirror line. Each point and its image are equidistant from the mirror line, on opposite sides.

For other mirror lines (e.g. x = 3 or y = −1), find the perpendicular distance from each point to the line and plot the image the same distance on the other side.

Rotation

A rotation turns a shape around a fixed centre by a specified angle in a specified direction.

Three pieces of information define a rotation completely:

• The centre of rotation (a coordinate)
• The angle of rotation (in degrees)
• The direction (clockwise or anticlockwise)

Omitting any one of these in an exam answer costs marks. A 180-degree rotation is the exception: direction is irrelevant because clockwise and anticlockwise produce the same image.

Worked example 2

Point A is at (3, 1). It is rotated 90 degrees anticlockwise about the origin. Find the image.

1. For a 90-degree anticlockwise rotation about (0, 0): (x, y) becomes (−y, x).
2. (3, 1) becomes (−1, 3).

Quick reference for rotations about the origin:

Enlargement

An enlargement changes the size of a shape by a scale factor from a fixed centre of enlargement.

To find the image of a point under enlargement with centre C and scale factor k:

1. Draw the vector from C to the original point.
2. Multiply that vector by k.
3. The image is at C plus the new vector.

Describing an enlargement

State three things: (1) that it is an enlargement, (2) the scale factor, and (3) the centre of enlargement. Missing any one loses marks.

Combined transformations

Exam questions sometimes ask for the single transformation equivalent to two successive transformations. The approach:

1. Apply the first transformation to the original shape to get an intermediate image.
2. Apply the second transformation to the intermediate image to get the final image.
3. Compare the original shape to the final image and describe the single transformation that maps one to the other.

Not every pair of transformations can be reduced to a single one of the four types. Two reflections in parallel lines produce a translation. Two reflections in intersecting lines produce a rotation. Be prepared to identify these patterns.

Vectors: the Extended tier

Everything below is Extended content. Core candidates can stop here.

Column vectors and notation

A vector has magnitude (length) and direction. In IGCSE Mathematics, vectors are written as column vectors or as bold lowercase letters (a, b) or with an arrow notation (AB with arrow above).

• The vector from A to B is written as the column vector (change in x, change in y).
• The magnitude of vector v = (p, q) is |v| = sqrt(p² + q²).
• A negative vector reverses direction: if AB = a, then BA = −a.

Vector arithmetic

Position vectors

The position vector of a point P is the vector from the origin O to P. If P = (5, −2), then the position vector of P is (5, −2).

The vector from A to B can be found using position vectors: AB = OB - OA. This is the most reliable method when coordinates are given.

Worked example 3

A has position vector (1, 3) and B has position vector (7, −1). Find the vector AB and the midpoint M of AB.

1. AB = OB - OA = (7, −1) - (1, 3) = (6, −4).
2. The midpoint M has position vector OM = OA + (1/2)AB = (1, 3) + (3, −2) = (4, 1).

Alternatively, M = ((1+7)/2, (3+(−1))/2) = (4, 1).

Expressing vectors in terms of base vectors

Many IGCSE vector questions provide a diagram where OA = a and OB = b, then ask you to express other vectors in terms of a and b. The method is always the same: trace a path from start to finish using known vectors.

Worked example 4

In triangle OAB, OA = a and OB = b. M is the midpoint of AB. Express OM in terms of a and b.

1. Find AB: AB = AO + OB = −a + b = b - a.
2. M is the midpoint of AB, so AM = (1/2)AB = (1/2)(b - a).
3. OM = OA + AM = a + (1/2)(b - a) = a + (1/2)b - (1/2)a = (1/2)a + (1/2)b.

So OM = (1/2)(a + b).

Proving collinearity

Three points X, Y, Z are collinear (lie on the same straight line) if XY is a scalar multiple of XZ. If XY = k(XZ) for some scalar k, then Y lies on the line through X and Z.

Worked example 5

OA = a, OB = b. Point P lies on OA such that OP = (2/3)a. Point Q lies on AB such that AQ = (2/3)AB. Show that O, Q, and some other relevant point are collinear, or find OQ.

1. AB = b - a.
2. AQ = (2/3)(b - a).
3. OQ = OA + AQ = a + (2/3)(b - a) = a + (2/3)b - (2/3)a = (1/3)a + (2/3)b.

If another point R has position vector that is a scalar multiple of (1/3)a + (2/3)b, then O, Q, and R are collinear.

Finding ratios

If P divides AB in the ratio m : n, then:

• AP = (m/(m+n)) AB
• OP = OA + AP = OA + (m/(m+n))(OB - OA)

This formula handles any ratio problem on the IGCSE paper.

Common mistakes

Self-check questions

1. A shape is translated by the vector (−4, 5). If one vertex is at (2, −3), where is its image?
2. Describe fully the single transformation that maps triangle A with vertices (1, 1), (3, 1), (3, 4) to triangle B with vertices (−1, 1), (−3, 1), (−3, 4).
3. A point (5, 2) is rotated 90° clockwise about the origin. State the coordinates of the image.
4. A shape is enlarged with scale factor 2 and centre (1, 1). The original point is (3, 4). Find the image.
5. If the scale factor of an enlargement is 0.5, what happens to the area of the shape?
6. Given a = (3, −1) and b = (−2, 5), find 2a + b and |2a + b|.
7. A has position vector (2, 6) and B has position vector (8, −2). Find the vector AB and the midpoint of AB.
8. In a triangle OAB, OA = a and OB = b. P is the point on AB such that AP:PB = 1:3. Express OP in terms of a and b.
9. Points X, Y, Z have position vectors (1, 2), (4, 5), and (7, 8). Show that X, Y, and Z are collinear.
10. A triangle is reflected in y = x, then reflected in y = −x. Describe the single transformation equivalent to these two reflections.