ECZ 2025 · O-Level Paper 2 · Question 12 — Geometrical Transformations

The vertices of a quadrilateral ABCD are A(1, 1), B(1, 3), C(3, 3) and D(4, 1) and the vertices of a quadrilateral A1B1C1D1 are A1(1, -2), B1(1, -6), C1(3, -6) and D1(4, -2).

(a) Using a scale of 1 cm to represent 1 unit on each axis for -9 ≤ x ≤ 6 and -9 ≤ y ≤ 6, draw and label quadrilaterals ABCD and A1B1C1D1.[2]

Verified working — step 1

Mark the scale on both axes first.

(b) Describe fully the transformation which maps quadrilateral ABCD onto quadrilateral A1B1C1D1.[3]

Verified working — step 1

Compare coordinates pair by pair: A(1, 1) → A1(1, -2), B(1, 3) → B1(1, -6), C(3, 3) → C1(3, -6), D(4, 1) → D1(4, -2).

(c) A rotation of 90° anti-clockwise about the origin maps quadrilateral ABCD onto quadrilateral A2B2C2D2.

(c(i)) Draw and label quadrilateral A2B2C2D2.[1]

Verified working — step 1

A quarter turn anticlockwise about the origin maps (x, y) → (-y, x).

(c(ii)) Find the matrix of this transformation.[2]

Verified working — step 1

A transformation matrix's columns are the images of (1, 0) and (0, 1).

(d) An enlargement maps quadrilateral ABCD onto quadrilateral A3B3C3D3 with vertices A3(-2, -2), B3(-2, -6), C3(-6, -6) and D3(-8, -2).

(d(i)) Draw and label quadrilateral A3B3C3D3.[1]

Verified working — step 1

Plot A3(-2, -2), B3(-2, -6…

(d(ii)) Find the centre and scale factor of the enlargement.[3]

Verified working — step 1

Compare object with image: A(1, 1) → (-2, -2), D(4, 1) → (-8, -2) — every coordinate is multiplied by -2.

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Original examination question © Examinations Council of Zambia. Worked solution and commentary © G12 Titan — not to be reproduced without permission.

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