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Given : In trapezium PQRS. side PQ || side SR, AR = 5AP, AS = 5AQ then prove that, SR = 5PQ

 

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Sol.       Side PQ || Side SR and PR is the transversal                        [Given]
             ∠QPR ≅ ∠SRP                              [P – A – R]
             ∠QPA ≅ ∠SRA                   ...(I)     [Converse of Corresponding angle test]
              Now Consider ΔPAQ and ΔRAS
              ∠QPA ≅ ∠SRA                              [From (I)]
              ∠PAQ ≅ ∠RAS                              [Vertically opposite angles]
∴            ΔPAQ ~ ΔRAS                              [AA test of similarity]

∴             \({AP\over AR} ={ AQ\over AS} = {PQ\over SR}\)             ...(II)     [Corresponding sides of similar triangles are proportional]

               AR = 5AP                                     [Given]

∴             \({AP\over AR} = {1\over 5}\)                          ...(III)

                AS = 5AQ                                    [Given]

∴             \({AQ\over AS} = {1\over 5}\)                          ...(IV)

                \({AP\over AR} ={ AQ\over AS} = {PQ\over SR}={1\over 5}\)                  [From (II), (III), (IV) and (V)]

∴               \({PQ\over SR} = {1\over 5}\)

                SR = 5 PQ

 

Similarity August 23 , 2018 0 Comments 18 views