Theory question 113
Constructions(a) Using a ruler and a pair of compasses only, construct: (i) (α) ΔABC with |AB| = 7.5 cm, |AC| = 13.5 cm and ∠ABC = 120°; (β) locus l1 of points equidistant from A and B; (γ) locus, l2 of points equidistant from B and C. (ii) Using N, the point of intersection of l1 and l2 as centre, draw a circle to pass through points A, B, and C.
(b) Using the method of completing the square, solve, 4x2 − 4√3x + 3 = 0, leaving the answer in surd form.
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(a)(i)(α) Draw AB = 7.5 cm; construct 120° at B (60° + 60°); with centre A, radius 13.5 cm, cut the arm at C (BC ≈ 8.1 cm)
(β) l1 = perpendicular bisector of AB; (γ) l2 = perpendicular bisector of BC
(ii) With centre N and radius NA (≈ 7.8 cm) draw the circle through A, B and C
(b) x2 − √3x = −¾
x2 − √3x + ¾ = 0 → (x − √3/2)2 = 0
x = √3/2 (twice)
Construction arcs must be left visible; the circle through A, B and C is the circumcircle.