Q. 34.3( 25 Votes )

Prove that a cyclic rhombus is a square.

Answer :

To prove rhombus inscribed in a circle is a square, we need to prove that either any one of its interior angles is equal to 90° or its diagonals are equal.



ABD = DBC = b


ADB = BDC = a


In the figure, diagonal BD is angular bisector of angle B and angle D.


In triangle ABD and BCD,


AD = BC (sides of rhombus are equal)


AB = CD (sides of rhombus are equal)


BD = BD (common side)


ABD BCD. (SSS congruency)


In the figure,


2a + 2b = 180° (as, opposite angles of a cyclic quadrilateral are always supplementary)


2(a + b) = 180°


a + b = 90°


In ABD,


Angle A = 180°-(a + b)


= 180°-90°


= 90°


Therefore, proved that one of its interior angle is 90°


Hence, rhombus inscribed in a circle is a square.


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