ECZ 2018 · GCE Paper 2 · Question 10 — Geometrical Transformations

Answer the whole of this question on a sheet of graph paper. Using a scale of 1 cm to represent 1 unit on both axes, draw x and y axes for -8 ≤ x ≤ 12 and -6 ≤ y ≤ 14.

(a) Draw and label triangle X with vertices (2, 4), (4, 4) and (4, 1).[1]

Verified working — step 1

Plot the vertices (2, 4), (4, 4)…

(b) Triangle X is mapped onto triangle U with vertices (6, 12), (12, 12) and (12, 3) by a single transformation.

(b(i)) Draw and label triangle U.[1]

Verified working — step 1

Plot the image vertices (6,…

(b(ii)) Describe fully this transformation.[3]

Verified working — step 1

Compare vertices: X(2, 4) → U(6, 12), X(4, 4) → U(12, 12), X(4, 1) → U(12, 3).

(c) A 90° clockwise rotation about the origin maps triangle X onto triangle W. Draw and label triangle W.[2]

Verified working — step 1

A 90° clockwise rotation about O sends (x, y) to (y, -x).

(d) A shear with x-axis as the invariant line and shear factor -2 maps triangle X onto triangle S. Draw and label triangle S.[2]

Verified working — step 1

A shear, x-axis invariant, factor -2, sends (x, y) to (x - 2y, y).

(e) Triangle X is mapped onto triangle M with vertices (4, 4), (8, 4) and (8, 1).

(e(i)) Draw and label triangle M.[1]

Verified working — step 1

Plot the image vertices (…

(e(ii)) Find the matrix which represents this transformation.[2]

Verified working — step 1

Compare X to M: (2, 4) → (4, 4), (4, 4) → (8, 4), (4, 1) → (8, 1).

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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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