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XY is a line parallel to side BC of a triangle ABC.If BE | | AC and CF || AB meet XY at Eand F respectively.Show that ar{ABE} = ar{ACF}

XY is a line parallel to side BC of a triangle ABC.If BE | | AC and CF || AB
 meet XY at Eand F respectively.Show that ar{ABE} = ar{ACF}

Grade:9

1 Answers

Arun
25763 Points
4 years ago
Dear Shashwat
 

Given : A triangle ABC. XY is parallel to BC; BE is parallel to AC and CY is parallel to AB.

To Prove : ar ( ABE ) = ar ( ACF ).

Proof : IIgm EBCY and triangle ABE being on the same base EB and between the same paralllels EB and CA, we have

ar ( ABE ) = 1/2 ar ( IIgm EBCY )  ...................... ( 1 )

Again, IIgm BCFX and triangle ACF being on the same base CF  and between the same parallels CF and BA, we have

ar ( ACF ) = 1/2 ar ( llgm BCFX )  ....................... ( 2 )

But, llgm EBCY and llgm BCFX being on the same base BC and between the same parallels BC and EF , we have

ar ( llgm EBCY ) = ar ( llgm BCFX )                                           ....................... ( 3 )

From,  ( 1 ), ( 2 ), and  ( 3 ) , we get

ar ( ABE ) = ar ( ACF )

 

Regards

Arun (askIITians forum expert)

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