`E , F , G , H` are respectively, the mid-points of the sides `A B ,B C ,C D` and `D A` of parallelogram `A B C D` . Show that the quadrilateral `E F G H` is a parallelogram and that its area is half the area of the parallelogram, `A B C Ddot` GIVEN : A quadrilateral `A B C D` in which `L ,F ,G ,H` are respectively the mid-points of the sides `A B ,B C ,C D` and `D Adot` TO PROVE : (i) Quadrilateral `E F G H` is a parallelogram `a r(^(gm)E F G H)=1/2a r(^(gm)A B C D)` CONSTRUCTION : Join `A C` and `H F`

Explanation

EF||AC and `EF=1/2AC-(1)`

In`/_ADC`, GH||AC and `GH=1/2AC-(2)`

EF=GH and EF||GH

EFGH is an parallelogram

ABCD is a parallelogram

AD=BC,AD||BC.

`1/2AD=1/2BC`

HD=FC,HD||FC

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