Question

ABCD is a quadrilateral where K,L,M and N are the mid points of AB,BC,CD and AD. BD is a diagonal, Show that : 1. KN II BD and KN = 1/2 BD 2. KN =LM 3.KNML is a parallelogram .​

Answer 1

Given: ABCD is a quadrilateral.

Points E,F,G,H are the midpoints of AB,BC,CD and DA.

Draw diagonals AC and BD in the quadrilateral ABCD

Segment HG is the midpoint segment in △ACD

∴ segment HG is parallel to the side AC of the

△ACD     (Line segment joining midpoints of two sides of a triangle property.)

Similarly, Segment EF is the midpoint segment in △ABC

∴ Segments EF is parallel to side AC of △ABC.

Since, Segment HG and EF are both parallel to the diagonal AC, they are parallel to each other.

Segment GF is the midpoint segment in △DCB.

∴ Segment GF is parallel to side DB of △ABD.

Segment HE is midpoint segment in △ABD

∴ Segment HE is parallel to side DB of triangle ABD

Since,Segment GF and HE are both parallel to diagonal DB, they are parallel to each other.

Thus, we have proved that in quadrilateral EFGH the opposite sides

HG and EF,HE and GF are parallel by pairs.

Hence, the quadrilateral EFGH is the parallelogram..