`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

Given: A quadrilateral `ABCD` in which `E`, `F`, `G`, `H` are respectively. The mid-points of the sides `AB`, `BC`, `CD` and `DA`.
Construction: Join `AC` and `HF`
`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`
`HDFC` is a parallelogram.
`ar(/_HGF)=1/2ar`(parallelogram `HDFC`) -(3)
Now,
`ar(/_HEF)=1/2ar`(parallelogram `HABF`) -(4)
Adding equation `3` and equation `4`
`ar(/_HGF)+ar(/_HEF)=1/2[ar(HDFC)+ar(HABF)]`
`ar(EFGH)=1/2ar(ABCD)`.