ECZ 2025 · GCE Paper 2 · Question 9 — Geometrical Transformations

Diagram shows on a grid: triangle PQR with P(1, 4), Q(2, 4), R(1, 6); triangle P1Q1R1 with P1(-4, 1), Q1(-4, 2), R1(-6, 1); triangle P2Q2R2 with P2(2, 8), Q2(4, 8), R2(2, 12); triangle P3Q3R3 with P3(-3, 4), Q3(-6, 4), R3(-3, 6). (Coordinates confirmed on the clean official booklet.)

(a) Describe fully a single transformation that maps triangle PQR onto triangle P1Q1R1.[3]

Verified working — step 1

Compare corresponding points: P(1,4)->P1(-4,1), Q(2,4)->Q1(-4,2), R(1,6)->R1(-6,1).

(b(i)) An enlargement maps triangle PQR onto triangle P2Q2R2. Find the centre of enlargement.[2]

Verified working — step 1

For an enlargement, lines joining object points to image points meet at the centre of enlargement.

(b(ii)) Find the scale factor.[1]

Verified working — step 1

Scale factor = image lengthobjectlength (or ratio of coordinates from centre).

(c) Triangle PQR is mapped onto triangle P3Q3R3 by a single transformation. Find the matrix of this transformation.[3]

Verified working — step 1

Let the matrix be abcd. For any point (x,y), image = (ax+by, cx+dy).

(d) The transformation with 1-201 maps triangle PQR onto triangle P4Q4R4, not shown on the diagram. Find the coordinates of triangle P4Q4R4.[3]

Verified working — step 1

Apply the 1-201 to each vertex using (x,y) -> (x-2y, y):

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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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