Question

show that the quadrilateral formed by joining the mid point of the pair of adjacent sides of rectangle is a rhombus

Answer 1

Let ABCD be the rectangle and E,F,G,H be the mid-points of AB,BC,CD and AD.Now since E,F,G,H are mid points AE=EB,AF=FC,CG=GD,AH=HD.
So, in triangle AEH and triangle GDH We have, 
GD= AE (as CG=DG=1/2CD=1/2AB(OPPOSITE SIDES OF A                                      RECTANGLE)=AE)
A=D (EACH=90 DEGREE)
AH=HD(SHOWN ABOVE)
BY SAS CONGRUENCY 
AEH CONGRUENT TO GDH 
By CPCT, EH=HG   -eq1
Similarly we can prove that CGF congruent to BEF.then by CPCT we get GF=EF -eq2
from eq1 and to we know that if adjacent sides are equal then its is rhombus