Theory question 108
ConstructionsUsing ruler and a pair of compasses only, (a) construct (i) a quadrilateral PQRS with |PS| = 6cm, ∠RSP = 90°, |RS| = 9cm, |QR| = 8.4cm and |PQ| = 5.4cm; (ii) the bisectors of ∠RSP and ∠SPQ to meet at X; (iii) the perpendicular XT to meet PS at T. (b) Measure |XT|.
Show the model answer
Model answer
(a)(i) Draw PS = 6 cm; construct 90° at S and mark SR = 9 cm. With centres P (radius 5.4 cm) and R (radius 8.4 cm) draw arcs meeting at Q outside triangle PSR; join PQ and QR.
(ii) Bisect ∠RSP (giving 45°) and ∠SPQ; X is where the bisectors meet.
(iii) From X drop a perpendicular to PS, meeting it at T.
(b) |XT| ≈ 3.4 cm.
Explanation
To drop a perpendicular from X, draw an arc from X cutting PS twice and bisect the segment between the cuts.