Show that the segment joining the mid-points of a pair of opposite sides of a parallelogram, divides it into two equal parallelograms. GIVEN : A parallelogram `A B C DdotE` and `F` are the mid-points of opposite sides `A B` and `C D` respectively. TO PROVE : `a r(^(gm)A E F D)=a r(^(gm)E B C F)` CONSTRCUTION : Join `E Fdot`

Explanation

`"ABCD is a parallelogram"`

`"AB parallel CD, AB=DC"`

`"AE parallel FD and "1/2 AB=1/2DC`

`"AE parallel FD and AE=FD"`

`therefore AEFD " is a parallelogram"`
`"Similarly EB=CF & EB//CF, BCEF is a parallelogram"`
`"Since parallelograms AEFD and BCEF lie on the same base and between same parallel lines"`

` ABCD" is a parallelogram"`

`therefore ar(AEFD)=ar(BCEF)`