Question

ABCD is a parallelogram. X and Y are the midpoints of the nonparallel sides AD and BC respectively prove that XY = 1/2(AB+CD)​

Answer 1

parallelogram and ab || cd. E and f are the mid points of bc and ad.

To prove,

1/2(ab + cd)

Construction,

Join be and produce it to meet cd produced at g.

Proof,

In ∆ edg and ∆ eab

Angle abe = angle egd.          [Alternate interior angles.]

De = ae         [ since, e is the mid point of ad.]

Angle aeb = angle ged       [vertically opposite angles.]

Therefore, ∆ edg congruent to ∆ eab  

By asa rule.

Ab = gd              [cpct]

Eb = eg              [cpct]

In ∆ cgb

Since, e is the mid point of eg [ since, ec  = eg]

Also, f is the mid point of bc.  [ Given]

Therefore, ef || gc and ef = 1/2 gc

By mid point theorem.

But, gc = gd + dc

Gc = ab + cd

That is, xy = 1/2( ab + cd )