# Cyclic Quadrilateral

## Trending Questions

**Q.**

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70∘, ∠BAC = 30∘, find ∠BCD. Further, if AB= BC, find ∠ECD.

**Q.**

If D is the mid-point of the hypotenuse AC of a right triangle ABC, prove that BD = 12AC. [4 MARKS]

**Q.**

In the figure, ∠PQR=100∘, where P, Q, and R are points on a circle with center O. Find ∠OPR.

20∘

10∘

30∘

25∘

**Q.**Question 13

A round table cover has six equal designs as shown in Fig. 12.14. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of ₹ 0.35 per cm2.(Use√3=1.7)

**Q.**

In the given figure, AOB is a diameter of a circle and CD||AB. If ∠BAD=30∘ then ∠CAD=?

(a) 30∘

(b) 60∘

(c) 45∘

(d) 50∘

**Q.**

ABCD is a cyclic quadrilateral in which BA and CD when produced meet in E and EA = ED. Prove that :

(i) AD || BC

(ii) EB = EC

**Q.**Question 11

ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠CAD=∠CBD.

**Q.**

In the figure, O is the centre of the circle. Find ∠CBD.

**Q.**

Question 19

In the figure, AOB is a diameter of the circle and C, D, E are any three points on the semi-circle. Find the value of ∠ACD+∠BED.

**Q.**

PQRS is a quadrilateral in which diagonals PR and QS intersect in O.

Find the correct relation between perimeter and diagonals of the quadrilateral.

PQ + QR + RS + SP < (PR + QS)

PQ + QR + RS + SP < 2 (PR + QS)

PQ + QR + RS + SP = (PR + QS)

PQ + QR + RS + SP > (PR + QS)

**Q.**

In the given figure, AOB is a diameter and ABCD is a cyclic quadrilateral. If ∠ADC=120∘ then ∠BAC=?

(a) 60∘

(b) 30∘

(c) 20∘

(d) 45∘

**Q.**

What is the sum of either pair of opposite angles of a cyclic quadrilateral?

90∘

180∘

360∘

45∘

**Q.**

Prove that the circles described on the four sides of a rhombus as diameter, pass through the point of intersection of its diagonals.

**Q.**

In the given figure, sides AB and AD of quad. ABCD are produced to E and F respectively. If ∠CBE=100∘ then ∠CDF=?

(a) 100∘

(b) 80∘

(c) 130∘

(d) 90∘

**Q.**

Prove that any exterior angle of a cyclic quadrilateral is equal to the inner angle at the opposite vertex.

**Q.**

Prove that the centre of the circle circumscribing the cyclic rectangle ABCD is the point of intersection of its diagonals.

**Q.**

In the figure △PQR is an isosceles triangle with PQ = PR and ∠PQR=35∘. Find ∠QTR.

- 40∘
- 70∘
- 50∘
- 60∘

**Q.**23. If the non parallel sides of a trapezium are equal, prove that it is cyclic?

**Q.**

Prove that the perpendicular bisectors of the sides of a cycle quadrilateral are concurrent.

**Q.**The sum of the angles in the four segments exterior to a cyclic quadrilateral is

360∘

270∘

540∘

180∘

**Q.**

ABCD is a rectangle. Prove that the centre of the circle through A, B, C, D is the point of intersection of its diagonals.

**Q.**Prove that the circle drawn with any side of a rhombus as diameter passes through the point of intersection of its diagonals.

**Q.**

Question 7

Write True or False and justify your answer :

ABCD is a cyclic quadrilateral such that ∠A = 90°, ∠B = 70°, ∠C = 95° and ∠D = 105°

**Q.**

**Question 8**

If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

**Q.**

If sum of opposite angles of a quadrilateral is 180∘, then the quadrilateral is a _____.

trapezium

cyclic quadrilateral

kite

parallelogram

**Q.**What is the difference between a cyclic and a concyclic quadrilateral?

**Q.**

Prove that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent.

**Q.**

**Question 3**

The angel between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 60∘. Find the angles of the parallelogram.

**Q.**Question 8

On a common hypotenuse AB, two right angled triangles, ACB and ADB are situated on opposite sides. Prove that ∠BAC=∠BDC

**Q.**In a triangleABC, AB=AC. Suppose the bisector of angleACBmeets AB at M. Let AM+MC=BC. Prove angleBAC= 100∘