Question

In a cyclic quadrilateral ABCD , if two sides are parallel , which of the following statement is definitely false ?
(a) Remaining two sides are equal
(b) Diagonals are not equal
(c) Diagonals intersect at the centre of the circle
(d) Both (a) & (c)​

Answer 1

Answer:

Let ABCD be the cyclic quadrilateral with AD parallel to BC. The center of the circle is O.

Join AC

Now, since AD is parallel to BC and AC is the transverse

angle DAC = angle ACB (interior opposite angles)

angle DAC is subtended by chord DC on the circumference and angle ACB is subtended by chord AB.

If the angles subtended by 2 chords on the circumference of the circle are equal, then the lengths of the chords is also equal.

Hence, DC = AB i.e. the 2 remaining sides of the cyclic quadrilateral are equal.

Now, BD and AC are the 2 diagonals.

In triangle ABD and triangle DCA

AB = CD (proved above)

angle DCA = angle DAB (angles subtended by the same chord AD)

angle ACB = angle DAC(proved above)

Hence, triangle ABD is congruent to triangle DCA (by ASA)

hence, BD = AC (by CPCT)

Answer 2

Answer:

Here's your answer!!!!

Step-by-step explanation:

In a cyclic quadrilateral if two sides are parallel, then remaining two sides are equal and diagonals are equal.

Hence,the false statement is (c) i.e. Diagonals intersect at the centre of the circle.

Hope it helps!!!!