Theory question 111
Constructions(a) Using a ruler and a pair of compasses only, construct a parallelogram PQRS with diagonals |PR| = 9cm and |QS| = 6cm intersecting at K and ∠QKR = 60°.
(b) Construct a rectangle PABS which is equal in area to PQRS in (a) above and on the same side of PS as PQRS. Measure |PA|
Show the model answer
(a) Draw PR = 9cm and bisect it to get K. Construct 60° at K for ∠QKR; mark KQ = KS = 3cm on the line through K. Join PQ, QR, RS, SP. Arcs must be shown.
(b) Area of PQRS = ½ × 9 × 6 × sin 60° ≈ 23.4cm2; PS ≈ 4.0cm
Construct perpendiculars to PS at P and S on the side of the parallelogram. The rectangle has the same base PS and lies between the same parallels, so the line QR (extended) cuts the perpendiculars at A and B.
|PA| ≈ 5.9cm (accept 5.8−6.0cm)
A rectangle and a parallelogram on the same base and between the same parallels have equal areas.