question_answer

If all the sides of a parallelogram touch a circle, then the parallelogram is a

A)  square             

B)  rectangle

C)  rhombus          

D)  trapezium

Correct Answer: C

Solution :

  ABCD is a parallelogram such that its sides touch a circle with centre 0. The tangents to a circle from an exterior point are equal in length. \[\therefore \] AP = AS; BP = BQ: CR = CQ and DR = DS. Adding all, AP + BP + CR + DR = AS + BQ + CQ+DS \[\Rightarrow \](AP + BP) + (CR + DR) \[\Rightarrow \] (AS + DS) + (BQ + CQ) \[\Rightarrow \]AB+CD=AD+BC \[\Rightarrow \]2AB = 2 BC \[\Rightarrow \] AB = BC \[\Rightarrow \]AB=BC=CD=AD Hence, ABCD is a rhombus.