Shape and Space 1
4 marks · HardThe diagram shows a quadrilateral PQRS with vertices at P(1,2), Q(5,2), R(4,6), and S(0,6).
Calculate the length of side PQ.
Show that PQRS is a parallelogram.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) 4 units · must show PQ = 4
(b) PQ||SR, PS||QR; opposite sides equal · must show PQ = SR = 4, PS = QR = 4.47 or \sqrt{20}, opposite sides parallel
Show mark scheme and worked solution
Marking points
M1 (a) · 0.5 marks · States or uses correct method to find side length PQ (x2-x1)
A1 (a) · 0.5 marks · Correct calculation and answer for PQ = 4
M2 (b) · 1 mark · Finds lengths of all sides, OR finds both pairs of opposite sides are parallel using gradients
M3 (b) · 1 mark · States both pairs of opposite sides are equal or parallel (evidence)
A2 (b) · 1 mark · Clearly concludes that PQRS is a parallelogram with justification
Worked solution
a) PQ is horizontal: PQ = x_2 - x_1 = 5 - 1 = 4 units
b) Calculate all sides: PQ = 4, QR = \sqrt{(4-5)^2 + (6-2)^2} = \sqrt{1 + 16} = \sqrt{17}\approx4.12
RS = |4-0| = 4, PS = \sqrt{(1-0)^2 + (2-6)^2} = \sqrt{1 + 16} = \sqrt{17}\approx4.12
Show sides PQ = RS and QR = PS. Gradients: PQ/RS both 0; QR/PS both undefined/vertical (parallel).
Therefore, PQRS has both pairs of opposite sides equal and parallel, so it is a parallelogram.