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Question
ABCD is a parallelogram. P and T are points on AB and DC respectively and AP = CT. Prove that PT and BD bisect each other.
Solution
Join AC
Since AC and BD are diagonals of a parallelogram, AC and BD bisect each other.
⇒ OA = OC and OD = OB ........(i)
AP = CT
But AB = CD
⇒ PB = DT
In ΔDOT and ΔPOB,
PB = DT
∠1 =∠2 ...(alternate angles since AB || CD)
∠3 = ∠4 ...(alternate angles since AB || CD)
Therefore, ΔDOT ≅ ΔPOB
Hence, OT = OP ........(ii)
From (i) and (ii)
OD = OB and OT = OP
Therefore, PT and BD bisect each other.
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