Mathematics
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.
Rectilinear Figures
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Answer
Square ABCD is shown in the figure below:
Considering △ABP and △ADQ we have,
⇒ ∠ABP = ∠DAQ = 90°
⇒ AP = DQ (Given)
⇒ AB = AD (Sides of square are equal)
Hence, △ABP ≅ △ADQ by RHS axiom.
⇒ ∠BAP = ∠ADQ (By C.P.C.T.) ……..(i)
⇒ ∠BAD = 90° (Each angle of square = 90°)
⇒ ∠BAP + ∠PAD = 90°
Substituting value of ∠ADQ from (i) we get,
⇒ ∠ADQ + ∠PAD = 90° ………(ii)
From figure,
∠ADQ = ∠ADM
∠PAD = ∠MAD
Substituting above values in (ii) we get,
⇒ ∠ADM + ∠MAD = 90° …….(iii)
In △AMD,
⇒ ∠ADM + ∠MAD + ∠AMD = 180°
⇒ 90° + ∠AMD = 180° (From iii)
⇒ ∠AMD = 90°
∴ AP ⊥ DQ.
Hence, proved that AP ⊥ DQ.
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