Question

prove that a cyclic rhombus is a square

Answer 1

For a cyclic quadrilateral, opposite angles are supplementary. This is possible only when it is 90 degree is a cyclic only if it is a rectangle. Rhombus: a rhombus is a cyclic only when it is a square..

Answer 2

Answer:

The proof is explained step-wise below :

Step-by-step explanation:

A rhombus is a quadrilateral with all sides of equal length.

But a square has not only all sides equal but also the measure of all interior angles are right angles.

So, to show : any rhombus is a square, we need to show any angle of a rhombus is right angle.

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

In ΔABC and ΔBCD,

AD = BC (sides of rhombus are equal)

AB = CD (sides of rhombus are equal)

BD = BD (common side)

△ABC ≅ △BCD. (SSS congruency postulate)

In the figure,

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

2(a + b)=180°  

a+b=90°

In △ABC,

⇒ ∠A = 180°- (a + b)

= 180°-90°

= 90°

Therefore,one of the interior angle of rhombus is 90°

Hence, rhombus inscribed in a circle is a square.

Hence Proved.