In a parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see the figure). Show that APCQ is a parallelogram.
Step-by-step solution
To find: Prove that APCQ is a parallelogram
Idea: Use the diagonal test: show that AC and PQ, the diagonals of APCQ, have the same midpoint O.
- Join AC; let it meet BD at O. The diagonals of parallelogram ABCD bisect each other: OA = OC and OB = OD.1 mark
- P and Q lie on opposite halves of BD (as in the figure). OP = OD − DP and OQ = OB − BQ. Since OD = OB and DP = BQ, OP = OQ.1 mark
- So in quadrilateral APCQ, the diagonals AC and PQ bisect each other at O. Hence APCQ is a parallelogram.1 mark
Check: With A(0, 0), B(6, 0), C(8, 4), D(2, 4) and DP = BQ = ¼BD: P(3, 3), Q(5, 1). Midpoint of AC = (4, 2) = midpoint of PQ ✓.
Answer to write in the exam
Join AC, meeting BD at O; OA = OC, OB = OD (diagonals of parallelogram ABCD bisect each other)
OP = OD − DP, OQ = OB − BQ; OD = OB and DP = BQ ⇒ OP = OQ
Diagonals AC and PQ of APCQ bisect each other at O
∴ APCQ is a parallelogram.
Common mistakes that cost marks
- Trying to prove triangles congruent without first using the midpoint O; the diagonal test is much shorter.
- Writing OP = OD + DP. P lies between D and O, so OP = OD − DP.
- Forgetting that the diagonals of APCQ are AC and PQ (not AP and CQ).
How this can come in the exam
ABCD is a parallelogram with diagonals meeting at O. E and F are points on AC with AE = CF. Prove that BEDF is a parallelogram.
Show answer
OA = OC and OB = OD (1 mark). OE = OA − AE = OC − CF = OF, so the diagonals BD and EF of BEDF bisect each other at O: BEDF is a parallelogram (1 mark).Try one yourself
In the question above, if BD = 20 cm and DP = BQ = 4 cm, find PQ.
Show answer
OD = OB = 10 cm, so OP = OQ = 6 cm and PQ = 12 cm.
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