ECZ 2023 · GCE Paper 2 · Question 4 — Construction and Loci
Construction on plain paper: triangle KLM.
(a(i)) Construct a triangle KLM in which KL = 10cm, LM = 7cm and angle KLM = 120°.[1]
Verified working — step 1
Draw line LM = 7cm. At L, use a protractor to measure an angle of 120° from LM and draw a ray. On this ray, mark off LK = …(a(ii)) Measure and write the length KM.[1]
Verified working — step 1
Using a ruler, measure the straight line KM on your accurate construction. KM ≈ 14.8 cm (this can be checked using the cosine rule: KM² = KL² + LM² − 2·KL·LM·cos120° = 100 + 49 − 2(10)(7)(−0.5) = 149 + 70 = 219, so KM = …(b(i)) Within triangle KLM, draw the locus of points 5.5cm from M.[1]
Verified working — step 1
Set compasses to a radius of 5.5cm. With the point on M, draw an ar…(b(ii)) Within triangle KLM, draw the locus of points 1cm from LM.[1]
Verified working — step 1
Draw a line parallel to LM, inside the triangle, at a perpendicular distance of 1cm from LM. This can be done by ma…(b(iii)) Within triangle KLM, draw the locus of points equidistant from M and L.[2]
Verified working — step 1
Construct the perpendicular bisector of ML: open compasses to more than half of ML, draw arcs from M and from L above and below the line so the…(c) A point P, within triangle KLM, is such that it is less than or equal to 5.5cm from M, greater than or equal to 1cm from LM and nearer to L than M. Indicate clearly, by shading, the region in which P must lie.[2]
Verified working — step 1
P must satisfy all three conditions at once: (1) inside or on the arc of radius 5.5cm from M (≤5.5cm from M), (2) on the far side of the 1cm-from-LM line away from LM (≥1cm from LM), (3) on the L-side of the perpendicular bisec…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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