Question

ABCD is a quadrilateral in which AD = BC. E, F, G and H are the mid-points of AB, BD, CD and Ac respectively. Prove that EFGH is a rhombus.

Answer 1

Given that AD = BC                                                    …..(1)

From the figure,
For triangle ADC and triangle ABD

2GH = AD and 2EF = AD, therefore 2GH = 2EF = AD …..(2)

For triangle BCD and triangle ABC

2GF = BC and 2EH=BC, therefore 2GF= 2EH = BC      …..(3)

From (1), (2) ,(3) we get,
2GH = 2EF = 2GF = 2EH
GH = EF = GF = EH
Therefore EFGH is a rhombus.
Hence proved.