Theory question 104
ConstructionsUsing ruler and a pair of compasses only, (a) construct ΔPQR such that |PQ| = 7cm, |PR| = 6cm and ∠PQR = 60° (b) Locate point M, the mid-point of PQ. (c) Measure ∠RMQ.
Show the model answer
Model answer
(a) Draw PQ = 7 cm. At Q construct 60° (equilateral-triangle arc). With centre P, radius 6 cm, draw an arc to meet the 60° arm at R.
Note: the distance from P to that arm is 7 sin 60° ≈ 6.06 cm, so a 6 cm arc only just touches it; R is at (or very near) the foot of the perpendicular from P, with QR ≈ 3.5 cm.
(b) Bisect PQ (perpendicular bisector) to get M, so QM = 3.5 cm.
(c) Triangle MQR has QM ≈ QR ≈ 3.5 cm and ∠Q = 60°, so it is (nearly) equilateral: ∠RMQ ≈ 60°.
Explanation
Construct 60° with compasses by drawing an equilateral triangle on the line.