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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:

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, ML Aggarwal Understanding Mathematics Solutions ICSE Class 9.

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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