InterviewSolution
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InterviewSolution
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In the adjoining figure, if points P, Q, R, S are on the sides of parallelogram such that AP = BQ = CR = DS, then prove that □PQRS is a parallelogram. |
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Answer» Given: □ABCD is a parallelogram. AP = BQ = CR = DS To prove: □PQRS is a parallelogram. Proof: □ABCD is a parallelogram. [Given] ∴ ∠B = ∠D ….(i) [Opposite angles of a parallelogram] Also, AB = CD [Opposite sides of a parallelogram] ∴ AP + BP = DR + CR [A-P-B, D-R-C] ∴ AP + BP = DR + AP [AP = CR] ∴ BP = DR ….(ii) In APBQ and ARDS, seg BP ≅ seg DR [From (ii)] ∠PBQ ≅ ∠RDS [From (i)] seg BQ ≅ seg DS [Given] ∴ ∆PBQ ≅ ∆RDS [SAS test] ∴ seg PQ ≅ seg RS …..(iii) [c.s.c.t] Similarly, we can prove that ∆PAS ≅ ∆RCQ ∴ seg PS ≅ seg RQ ….(iv) [c.s.c.t] From (iii) and (iv), □PQRS is a parallelogram. [A quadrilateral is a parallelogram, if pairs of its opposite angles are congruent] |
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