ECZ 2012 · O-Level Paper 2 · Question 12 — Geometrical Transformations
Using a scale of 2 cm to represent 10 units on each axis, draw x and y axes for -30 ≤ x ≤ 40 and -30 ≤ y ≤ 30.
(a) Triangle ABC has vertices A(-30, 10), B(-30, 20) and C(-10, 20). Triangle A1B1C1 has vertices A1(10, -30), B1(20, -30) and C1(20, -10).
(a(i)) Draw and label triangle ABC and triangle A1B1C1.[1]
Verified working — step 1
Plot triangle ABC at (-30, 10), (-30, 20), (-10, 20) and join.(a(ii)) Describe fully a single transformation that maps triangle ABC onto triangle A1B1C1.[2]
Verified working — step 1
Compare vertices: A(-30, 10) → (10, -30), B(-30, 20) → (20, -30), C(-10, 20) → (20, -10).(b) An enlargement centre (10, -30) and scale factor 2 maps triangle A1B1C1 onto triangle A2B2C2. Draw and label triangle A2B2C2.[1]
Verified working — step 1
A1 is the centre of enlargement, so A2 = A1 = (10, -30).(c) A reflection in the line y = 0 maps triangle ABC onto triangle A3B3C3. Draw and label triangle A3B3C3.[1]
Verified working — step 1
Reflection in y = 0 (the x-axis) sends (x, y) to (x, -y).(d(i)) Triangle A4B4C4 is the image of triangle A1B1C1 under a translation. Given that A4 is the point (10, 10), write the column vector representing this translation.[2]
Verified working — step 1
A1 is (10, -30) and A4 is (10, 10).(d(ii)) draw and label triangle A4B4C4.[1]
Verified working — step 1
Add 040 to each vertex of A1B1C1.(e) The matrix 11201 maps triangle A4B4C4 onto triangle A5B5C5.
(e(i)) Draw and label triangle A5B5C5.[2]
Verified working — step 1
The matrix sends (x, y) to (x + y2, y).(e(ii)) Describe fully the single transformation represented by this matrix.[2]
Verified working — step 1
Points with y = 0 do not move, so the x-axis is the invariant line.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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