ECZ 2019 · GCE Paper 2 · Question 11 — Geometrical Transformations
Diagram shows triangles ABC, A1B1C1, A2B2C2 and A3B3C3 on a grid (coordinates NOT printed in text — diagram-dependent). Both answer scripts read the object triangle as A(2, 2), B(2, 0), C(0, 1).
(a(i)) An enlargement maps triangle ABC onto triangle A1B1C1. Find the centre of enlargement.[1]
Verified working — step 1
Join each vertex of ABC to its image on A1B1C1 and extend: A(2,2) to A1(-4,-4), B(2,0) to B1(-4,0), C(0,1) to C1(0,-2). All three lines pass through the same point, the origin.(a(ii)) Find the scale factor.[1]
Verified working — step 1
The image is on the opposite side of the centre and each image point is twice as far from the centre as its object point: A(2,2) → A1(-4,-4) = -2 × (2,2).(b) Triangle ABC is mapped onto triangle A2B2C2 by a single transformation. Describe fully this transformation.[3]
Verified working — step 1
Compare corresponding points: A(2,2) → A2(2,-2), B(2,0) → B2(0,-2), C(0,1) → C2(1,0). Each point (x, y) maps to (y, -x), which is the rule for a rotation of 90° clockwise about the origin. Lengths are unchanged, so it is a rotation, not an enlargement.(c(i)) Triangle ABC is mapped onto triangle A3B3C3 by a stretch. Find the matrix which represents this transformation.[3]
Verified working — step 1
A stretch keeps an invariant line fixed and multiplies distances from it by a constant factor. Comparing ABC and A3B3C3: C(0,1) → C3(0,1) is unchanged (it lies on the y-axis), B(2,0) → B3(-4,0) and A(2,2) → A3(-4,2): y-coordinates are unchanged and x-coordinates are multiplied by -2. So this is a stretch with factor -2, invariant line the y-axis.(c(ii)) Find the area scale factor.[1]
Verified working — step 1
Area scale factor = |determinant of the stretch matrix| = |(-2)(1) - (0)(0)| = …(d) A transformation 1021 maps triangle ABC onto triangle A4B4C4 (not drawn). Find the coordinates of A4, B4 and C4.[3]
Verified working — step 1
Apply the 1021 to each vertex of ABC: x' = x and y' = 2x + y.The full step-by-step working and final answer are included with Premium — checked against ECZ marking standards.
Original examination question © Examinations Council of Zambia. Worked solution and commentary © G12 Titan — not to be reproduced without permission.
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