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Question
ABCD is a square. A is joined to a point P on BC and D is joined to a point Q on AB. If AP = DQ;
prove that AP and DQ are perpendicular to each other.
Solution
ABCD is a square and AP = PQ
Consider ΔDAQ and ΔABP,
∠DAQ = ∠ABP = 90°
DQ = AP
AD = AB
ΔDAQ ≅ ΔABP
⇒ ∠PAB = ∠QDA
Now,
∠PAB + ∠APB = 90°
also ∠QDA + ∠APB = 90° ...[∠PAB = ∠QDA]
Consider ΔAOQ by ASP
∠QDA + ∠APB + ∠AOD = 180°
⇒ 90° + ∠AOD = 180°
⇒ ∠AOD = 90°
Hence, AP and DQ are perpendicular.
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