Cyclic Quadrilateral

Q.

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If $\angle$DBC = 70${}^{\circ }$, $\angle$BAC = 30${}^{\circ }$, find $\angle$​BCD. Further, if AB= BC, find $\angle$​ECD.

Q.

If D is the mid-point of the hypotenuse AC of a right triangle ABC, prove that BD = $\frac{1}{2}$AC. [4 MARKS]

Q.

In the figure, $\angle PQR$$={100}^{\circ }$, where $P,Q,$ and $R$ are points on a circle with center $O.$ Find $\angle OPR.$

1. ${20}^{\circ }$

2. ${10}^{\circ }$

3. ${30}^{\circ }$

4. ${25}^{\circ }$

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 $c{m}^2.(Use\sqrt{3}=1.7)$

Q.

In the given figure, AOB is a diameter of a circle and CD||AB. If $\angle BAD={30}^{\circ }then\angle CAD=?$

(a) ${30}^{\circ }$

(b) ${60}^{\circ }$

(c) ${45}^{\circ }$

(d) ${50}^{\circ }$

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 $\angle CAD=\angle CBD$.

Q.

In the figure, O is the centre of the circle. Find $\angle CBD$.

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 $\angle ACD+\angle BED$.

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.

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

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

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

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

Q.

In the given figure, AOB is a diameter and ABCD is a cyclic quadrilateral. If $\angle ADC={120}^{\circ }\text{then}\angle BAC=?$

(a) ${60}^{\circ }$

(b) ${30}^{\circ }$

(c) ${20}^{\circ }$

(d) ${45}^{\circ }$

Q.

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

1. ${90}^{\circ }$

2. ${180}^{\circ }$

3. ${360}^{\circ }$

4. ${45}^{\circ }$

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 $\angle CBE={100}^{\circ }then\angle CDF=?$

(a) ${100}^{\circ }$

(b) ${80}^{\circ }$

(c) ${130}^{\circ }$

(d) ${90}^{\circ }$

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 $\angle PQR={35}^{\circ }$. Find $\angle QTR$.

1. ${40}^{\circ }$
2. ${70}^{\circ }$
3. ${50}^{\circ }$
4. ${60}^{\circ }$

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

1. 
   ${360}^{\circ }$
2. 
   ${270}^{\circ }$
3. 
   ${540}^{\circ }$
4. 
   ${180}^{\circ }$

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 ${180}^{\circ }$, then the quadrilateral is a _____.

1. trapezium

2. cyclic quadrilateral

3. kite

4. 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 ${60}^{\circ }$. 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 $\angle BAC=\angle BDC$

Q. In a triangleABC, AB=AC. Suppose the bisector of angleACBmeets AB at M. Let AM+MC=BC. Prove angleBAC= ${100}^{\circ }$