Question

9. In a parallelogram ABCD, mark two points E and F on the diagonal AC on either side of O (the point of intersection of the diagonals AC and BD) such that AE = CF. Prove that BEDF is a parallelogram.
[HOTS]​

Answer 1

Given: E and F are points on diagonal AC of parallelogram ABCD with AE=CF

To prove: $BFDE$ is a parallelogram

Construction: Join  $DE, DF, BE, BF, DB$

 

$\huge \sf {\purple{\underline {\pink{\underline { proof: }}}}$

∵ diagonals of a ║gm bisect each other

⇒ $OA = OC$

⭢ $OD = OB$

⭢∵$OA = OC and AE = CF (Given)$

⭢ $OA-AE = OC-CF$

⭢$OE = OF$

∵$BFDE$ is a quadrilateral whose diagonals bisect each other

Answer 2

Step-by-step explanation:

∵ diagonals of a ║gm bisect each other

⇒ OA = OCOA=OC

⭢ OD = OBOD=OB

⭢∵OA = OC and AE = CF (Given)OA=OCandAE=CF(Given)

⭢ OA-AE = OC-CFOA−AE=OC−CF

⭢OE = OFOE=OF

∵BFDEBFDE is a quadrilateral whose diagonals bisect each other