Theory question 109
ConstructionsUsing a ruler and a pair of compasses only. (a) Construct: (i) ΔPQR such that |PQ| = 8cm, |PR| = 7cm and ∠QPR = 105°, (ii) locus l1 of points equidistant from P and Q. (iii) locus l2 of points equidistant from Q and R (b) (i) Label the point T and where l1 and l2 intersect. (ii) With centre T and radius |TQ|, construct a circle l3. (iii) Complete quadrilateral PQSR such that |RS| = |QS| and |TQ| = |TS|.
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(a)(i) Draw PQ = 8 cm. Construct 105° at P (90° plus half of the 30° between 90° and 120°). Mark R on this arm with PR = 7 cm; join QR (QR ≈ 11.9 cm).
(ii) l1: perpendicular bisector of PQ.
(iii) l2: perpendicular bisector of QR.
(b)(i) T is where l1 and l2 meet (the circumcentre).
(ii) The circle centre T radius TQ (≈ 6.2 cm) passes through P, Q and R.
(iii) S is where l2 meets the circle on the side of QR away from P; join QS and RS.
TS = TQ puts S on the circle, and RS = QS puts S on the perpendicular bisector of QR.