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TMUA 2019 · Paper 2 · Question 10 of 20

TMUA 2019 Paper 2 Question 10

Coordinate geometry — Parallelograms · necessary but not sufficient. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 2Coordinate geometryParallelograms · necessary but not sufficient4 options
PQRS is a quadrilateral, labelled anticlockwise.

Which one of the following is a necessary but not sufficient condition for PQRS to be a parallelogram?

  1. APQ = SR and PS is parallel to QR
  2. BPQ = SR and PQ is parallel to SR
  3. CPQ = QR = SR = PS
  4. DPR = QS
Show the answer and worked solution
answer · A
  1. APQ = SR and PS is parallel to QR
  2. BPQ = SR and PQ is parallel to SR
  3. CPQ = QR = SR = PS
  4. DPR = QS
In the parallelogram PQRS the opposite sides are PQ with SR, and QR with PS; both pairs are equal and parallel. The first option asks for one pair equal and the other pair parallel, which every parallelogram satisfies, so it is necessary — but an isosceles trapezium with PS parallel to QR and equal slanted sides PQ = SR satisfies it too, so it is not sufficient. The second option pairs equality and parallelism on the same side, which is the standard sufficient test, so it is both. The rhombus condition is sufficient but not necessary, and equal diagonals is neither.