ECZ 2024 · GCE Paper 2 · Question 8 — Geometrical Transformations

Diagram shows on a grid: triangle ABC with A(2, -2), B(4, -2), C(4, -4); triangle GHI with G(-2, -2), H(-2, -4), I(-4, -4); triangle PQR with P(-4, 4), Q(-8, 4), R(-8, 8); triangle KLM with K(2, 2), L(4, 6), M(4, 4). (Coordinate reads agreed identically by both external scripts; every transformation recomputes exactly.)

(a) Describe fully the single transformation which maps triangle ABC onto triangle GHI.[3]

Verified working — step 1

Compare coordinates: A(2,-2)->G(-2,-2), B(4,-2)->H(-2,-4), C(4,-4)->I(-4,-4).

(b) Triangle ABC is mapped onto triangle PQR by an enlargement. Find the matrix representing this transformation.[3]

Verified working — step 1

Let the enlargement matrix be a00d since axes map to axes (no rotation).

(c) A stretch represented by the 1002 maps triangle ABC onto triangle DEF (not in the diagram). Find the coordinates of D, E and F.[3]

Verified working — step 1

Apply the 1002 to each vertex of ABC:

(d) Triangle KLM is the image of triangle ABC under a shear. Find the shear factor and invariant line.[3]

Verified working — step 1

Since the x-coordinates of A,B,C stay the same in K,L,M, the shear is parallel to the y-axis, so the invariant line is the y-axis (x = 0).

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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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