IGCSE Mathematics: Transformations and Vectors — Practice Questions

These are original questions written for Marlbridge, in the style and at the standard of the examination. They are not reproduced past-paper questions — examination boards hold copyright in their own papers. Use these alongside the official past papers available free from your board.

Related: Transformations and Vectors revision notes

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Questions

1. Triangle A has vertices (1, 1), (3, 1) and (1, 4).

(a) Describe fully the single transformation that maps triangle A onto triangle B, where B has vertices (1, −1), (3, −1) and (1, −4). [2] (b) Triangle A is enlarged by scale factor 3, centre the origin. State the coordinates of the image of the point (1, 4). [2]

2. Describe fully the single transformation that maps a shape onto its image after a rotation of 180° about the point (2, 3), given that this is what has been applied. State the two pieces of information (other than “rotation”) that must always be given to describe a rotation fully. [3]

3. (Extended) a = (3, 4) and b = (−6, 1) are vectors.

(a) Find a + b. [1] (b) Find 2a − b. [2]

4. (Extended) Find the magnitude of the vector v = (−5, 12). [2]

5. (Extended) In triangle OAB, OA = a and OB = b. M is the midpoint of AB.

(a) Write OM in terms of a and b. [2] (b) P is the point such that OP = 2 × OM. Write OP in terms of a and b, and show that OAPB is a parallelogram. [3]

6. Triangle A has vertices (1, 1), (3, 1) and (1, 4). Triangle D has vertices (5, −1), (7, −1) and (5, 2), formed by translating every vertex of triangle A by the same column vector. Describe fully the single transformation that maps triangle A onto triangle D. [2]

7. Given OM = a and ON = b, write down the vector MN in terms of a and b, and state how the vector NM relates to it. [2]

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Answers

1. (a) Reflection in the x-axis (the line y = 0) [2]. (b) Enlargement scale factor 3 about the origin multiplies each coordinate by 3: (1, 4) → (3, 12) [2].

2. A rotation must always state the angle of rotation (with direction, if not 180°) and the centre of rotation [2]. Here: rotation of 180° about (2, 3) [1].

3. (a) a + b = (3 + (−6), 4 + 1) = (−3, 5) [1]. (b) 2a − b = (2×3 − (−6), 2×4 − 1) = (6 + 6, 8 − 1) = (12, 7) [2].

4. |v| = √((−5)² + 12²) [1] = √(25 + 144) = √169 = 13 [1].

5. (a) OM = OA + ½AB = a + ½(b − a) [1] = ½a + ½b (or ½(a + b)) [1]. (b) OP = 2 × ½(a + b) = a + b [1]. AP = OP − OA = (a + b) − a = b = OB [1]. Since AP = OB, AP is parallel and equal in length to OB, so OA is parallel and equal to BP — both pairs of opposite sides are parallel and equal, so OAPB is a parallelogram [1].

6. Column vector = (image x-coordinate − object x-coordinate, image y-coordinate − object y-coordinate) using the first vertex: (5 − 1, −1 − 1) = (4, −2) [1]. Translation by the column vector (4, −2) [1].

7. MN = ON − OM = b − a [1]. NM is the vector in the opposite direction to MN, so NM = OM − ON = a − b, the negative of MN [1].

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Where marks are usually lost

• Describing a transformation incompletely — a reflection needs the mirror line stated, a rotation needs both the angle and the centre, an enlargement needs both the scale factor and the centre, and a translation needs the full column vector.
• For an enlargement about the origin, forgetting to multiply both coordinates by the scale factor, or applying it as an addition instead of a multiplication.
• Subtracting vector components in the wrong order when finding a vector between two points (it is “end point minus start point”).
• Forgetting to square-root at the end when finding a vector’s magnitude, or mixing up the magnitude formula with the midpoint formula.
• In vector-geometry proofs, writing a route like OM as OA + AM without also converting AM into a and b terms — every step of a vector route must end up expressed only in the given vectors (here, a and b).

Examiner report insight

• To show three points are collinear, compare the two relevant displacement vectors (for example the route between the first and second point, and the route between the first and third) and show one is a scalar multiple of the other – their shared direction can itself be a combination of both given vectors (for example b − a), so seeing both given vectors in the simplified answer is not by itself a sign of an error; what matters is that the two compared vectors are proportional.
• Vector direction matters: the vector from M to N is the negative of the vector from N to M – check carefully which one a question actually asks for, especially after working out a route in the opposite direction.

Source: Cambridge International, 0580 Mathematics Principal Examiner Report, June 2024 series, Papers 21, 23 (verified 2026-09-02).

Approaching transformations and vectors questions

For a “describe fully” transformation question, write down each required piece of information as a separate labelled item before combining it into a sentence – mirror line, or angle and centre, or scale factor and centre, or column vector – since a correct transformation named without its required extra detail typically only earns partial credit. For vector-geometry routes, treat every unconverted intermediate vector as unfinished working rather than a final answer, and keep track of direction carefully: the vector from one named point to another is always end point minus start point, and reversing that order simply negates the result rather than producing a different, unrelated vector.