ECZ 2018 · GCE Paper 2 · Question 10 — Geometrical Transformations
Answer on graph paper (1cm to 1 unit, -8 ≤ x ≤ 12, -6 ≤ y ≤ 14). Triangle X has vertices (2,4), (4,4), (4,1). Triangle U has vertices (6,12), (12,12), (12,3). Triangle M has vertices (4,4), (8,4), (8,1).
(a) Draw and label triangle X.[1]
Verified working — step 1
Plot points (2,4), (4,4), (4,1) on the grid using 1cm = …(b(i)) Draw and label triangle U.[1]
Verified working — step 1
Plot points (6,12), (12,12), (12,3) on the grid. Join them to fo…(b(ii)) Describe fully the single transformation that maps X onto U.[3]
Verified working — step 1
Compare corresponding vertices: X(2,4)->U(6,12), X(4,4)->U(12,12), X(4,1)->U(12,3). Each coordinate of U is 3 times the corresponding coordinate of X, so the scale factor k = …(c) A 90° clockwise rotation about the origin maps X onto W. Draw and label triangle W.[2]
Verified working — step 1
A 90° clockwise rotation about the origin maps (x,y) -> (y,-x). Apply to each vertex of X: (2,…(d) A shear with x-axis invariant and shear factor -2 maps X onto S. Draw and label triangle S.[2]
Verified working — step 1
A shear with the x-axis invariant and shear factor k=-2 maps (x,y) -> (x+ky, y) = (x-2y, y). Apply to each vertex of X: (2,4)->(2-8,4)=(-6,4); (4,4)->(4-8,4)=(-4,4); (4,1)->(4-2,1) = …(e(i)) Draw and label triangle M.[1]
Verified working — step 1
Plot points (4,4), (8,4), (8,1) on the grid. Join the…(e(ii)) Find the matrix which represents the transformation X onto M.[2]
Verified working — step 1
We need a abcd such that it maps X's vertices to M's vertices. Using vertex (2,4)->(4,4) and (4,4)->(8,4), we see x-coordinates double while y stays the same: (x,y)->(2x,y). Check with (4,1)->(8,1): 2(4)=8, y=1, correct. So the matrix is 2001, since 2001xy = …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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