Question

Which of the following is a cyclic quadrilateral?
(This question has multiple correct options)
(a). Rhombus
(b). Rectangle
(c). Parallelogram
(D). Trapezium

Answer 1

Hint: Before solving this question, we must know about cyclic quadrilaterals. They are explained below:-
CYCLIC QUADRILATERALS: A cyclic quadrilateral is any four – sided figure whose all four vertices lie on a circle.
 Also, in a cyclic quadrilateral, the sum of each pair of opposite angles is 180°. This means that if a quadrilateral has one pair of opposite angles that add to 180°, then you know it is cyclic.

Complete step-by-step solution -
Let us now solve the question.
We shall consider every option.
Rhombus: A rhombus cannot be a cyclic quadrilateral because, as mentioned in the hint provided above, the opposite angles of a cyclic quadrilateral are supplementary, but in the case of a rhombus, the opposite angles are equal. So, it cannot be a cyclic quadrilateral.
Rectangle: Every rectangle, including the special case of a square, is a cyclic quadrilateral because a circle can be drawn around it touching all four vertices and, also, the opposite angles of a rectangle are supplementary, i.e. they add up to make 180°. Hence, it is a cyclic quadrilateral.
Parallelogram: Any parallelogram cannot be cyclic because if any quadrilateral is cyclic, then the sum of the opposite angles must be 180°. But in the case of a parallelogram, the opposite sides are equal, not supplementary. Therefore, it cannot be a cyclic quadrilateral.
Trapezium: A cyclic quadrilateral is any four – sided figure whose all four vertices lie on a circle. A trapezium is cyclic only if it is an isosceles trapezium.
As mentioned above, this question has multiple correct options; therefore, the answer of this question can be rectangle as well as trapezium, only if the trapezium is an isosceles trapezium.
Therefore, the correct options are (b) Rectangle and (d) Trapezium, if the trapezium is isosceles.
NOTE:-
Let us now know about the other properties of a cyclic quadrilateral.
The opposite angles of a cyclic quadrilateral are supplementary.
If the sum of the two opposite angles is 180°, then it’s a cyclic quadrilateral.
If PQRS is a cyclic quadrilateral, then ∠SPR = ∠SQR, ∠QPR = ∠QSR, ∠PQS = ∠PRS, ∠QRP = ∠QSP.
The four vertices of a cyclic quadrilateral lie on the circumference of the circle.