Angles Subtended by an Arc at the Centre Is Twice the Angle Subtended at the Circumference

Q. A ladder, leaning against a wall, makes an angle of ${60}^{\circ }$ with the horizontal. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder.

Q. Question 13
Diagonals of a quadrilateral ABCD bisect each other. If $\angle A={35}^{\circ }$, determine $\angle B.$

Q.

In the following figure, O is centre of the circle, $\angle AOB={60}^{\circ }$ and $\angle BDC={100}^{\circ }$, Find $\angle OBC$

1. $\angle OBC={50}^{\circ }$

2. $\angle OBC={90}^{\circ }$

3. $\angle OBC={10}^{\circ }$

4. $\angle OBC={120}^{\circ }$

Q. Lines RS and PQ intersect each other at point O. If $\angle POR:\angle ROQ$ = 11:13, $\angle SOQ$ =

1. ${82.5}^{\circ }$
2. ${87.5}^{\circ }$
3. ${92}^{\circ }$

Q. Question 97
State whether the statements are True or False.
Sum of all the angles of a quadrilateral is ${180}^{\circ }$.

Q.

ABC is the right angled triangle with $\angle$ABC $={90}^{\circ }$, The centre of the circle passing through ABC lies on _______.

1. AC

2. AB

3. BC

4. a circle cannot be drawn through passing through A, B, C

Q. Question 6
In the figure, POQ is a line. The value of x is:


A) ${20}^{\circ }$
B) ${25}^{\circ }$
C) ${30}^{\circ }$
​​​​​​​D) ${35}^{\circ }$

Q.

A rod bent into an angle is placed with its vertexr at the centre of a circle and it is found that $\frac{1}{10}$ of the circle lies within it. If it is placed with its vertex on the circumference, what part of the circle would be within it ?

1. $\frac{1}{10}$

2. $\frac{1}{6}$

3. $\frac{1}{5}$

4. $\frac{3}{10}$

Q. Question 3
For which of the following, all angles are equal?
a) Rectangle
b) Kite
c) Trapezium
d) Rhombus

Q.

In the given figure, the central angle of arc AXB is 40 and the central angle of arc CYD is ${70}^{\circ }$.

Find the angles $\angle CAD$ and $\angle PDA$.

1. $\angle CAD={35}^o,\angle PDA={20}^o$
2. $\angle CAD={45}^o,\angle PDA={20}^o$
3. $\angle CAD={35}^o,\angle PDA={40}^o$
4. $\angle CAD={45}^o,\angle PDA={40}^o$

Q.

Three vertices of a parallelogram are on a circle and the fourth vertex is at the center. Find $\angle$ ABC.

1. ${110}^0$

2. ${120}^0$

3. ${60}^0$

4. ${100}^0$

Q.

In the given figure, AB and CD are diameters of the circle and AC=4 cm and $\angle DPB={45}^{\circ }$

Find the radius if the circle.

1. $3\sqrt{2}$ cm

2. $7\sqrt{2}$ cm

3. $2\sqrt{2}$ cm

4. $4\sqrt{2}$ cm

Q.

In the given figure, O is the centre of the circle and A, B, C

are points on it, then $\angle OAC+\angle ABC=$___.

1. ${70}^{\circ }$

2. ${90}^{\circ }$

3. ${120}^{\circ }$

4. ${60}^{\circ }$

Q.

If three vertices of a parallelogram are on a circle and fourth vertex is at the centre, then find the value of x.

1. ${70}^{\circ }$
2. ${60}^{\circ }$
3. ${50}^{\circ }$
4. ${100}^{\circ }$

Q.

If $\angle POR=120°$, then the value of $\angle PQR=$ .

1. 130°
2. 60°
3. 90°
4. 120°

Q. In the given figure, AB is the chord to the circle having centre at O. If AC = CB and $\angle AOC={50}^{\circ }$, then $\angle CAO$ is equal to.

1. ${30}^{\circ }$
2. ${50}^{\circ }$
3. ${40}^{\circ }$

Q.

Given a circle with O as the centre. Find the value of x.

1. 80°

2. 150°

3. 50°

4. 40°

Q.

ABC is a right angled triangle with $\angle$ABC=${90}^{\circ }$, the centre of the circle passing through A, B, C lies on _______.

1. AB

2. AC

3. BC

4. a circle cannot be drawn through passing through A, B, C

Q. In the given figure, A, B and C are three points on a circle with centre O such that $\angle BOC={30}^{\circ }$ and $\angle AOB={60}^{\circ }$. If D is a point on the circle other than the arc ABC, find $\angle ADC.$

1. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${50}^{\circ }$
4. ${60}^{\circ }$

Q.

Calculate the measure of $\angle AOC.$

1. ${120}^{\circ }$

2. ${70}^{\circ }$

3. ${140}^{\circ }$

4. ${90}^{\circ }$

Q. In a circle of radius $21cm$, an arc subtends and angle of ${60}^{\circ }$ at the centre. Find:
(i) The length of the arc (ii) Area of the sector formed by the arc (iii) Area of the segment formed by the corresponding chord

Q.

BC is chord of a circle with centre O. A is a point on major arc BC. Find the total measure of $\angle BAC$ and $\angle OBC$.

1. ${100}^{\circ }$

2. ${90}^{\circ }$

3. ${120}^{\circ }$

4. ${150}^{\circ }$

Q. In the given figure, A, B and C are three points on a circle with centre O such that $\angle BOC={30}^{\circ }$ and $\angle AOB={60}^{\circ }$. If D is a point on the circle other than the arc ABC, find $\angle ADC$.

1. $(\frac{90}{2}{)}^{\circ }$
2. ${45}^{\circ }$
3. $(\frac{60}{2}{)}^{\circ }$
4. ${30}^{\circ }$

Q. Question 38
Which of the following can be four interior angles of a quadrilateral?
a) ${140}^{\circ },{40}^{\circ },{20}^{\circ },{160}^{\circ }$
b) ${270}^{\circ },{150}^{\circ },{30}^{\circ },{20}^{\circ }$
c) ${40}^{\circ },{70}^{\circ },{90}^{\circ },{60}^{\circ }$
d) ${110}^{\circ },{40}^{\circ },{30}^{\circ },{180}^{\circ }$

Q. If the area of a circle inscribed in an equilateral triangle is $4\pi c{m}^2$. then what is the area of the triangle?

1. $12\sqrt{3}c{m}^2$
2. $9\sqrt{3}c{m}^2$
3. $8\sqrt{3}c{m}^2$
4. $18c{m}^2$

Q.

The measure of $\angle AOC=$ ___

1. ${130}^{\circ }$

2. ${120}^{\circ }$

3. ${100}^{\circ }$

4. ${140}^{\circ }$

Q.

An equilateral triangle ABC is inscribed in a circle with centre O. Then, $\angle BOC$ is equal to ___.

1. 30${}^{\circ }$
2. 60${}^{\circ }$
3. 90${}^{\circ }$
4. 120${}^{\circ }$

Q.

The number of equal angles an arc subtends in the opposite segment is two.

1. True

2. False

Q. A chord of a circle of radius $15cm$ subtends an angle of ${60}^{\circ }$ at the centre. Find the areas of the corresponding minor and major segments of the circle.
(Use $\pi =3.14$ and $\sqrt{3}=1.73$)

Q. In the given figure, O is the centre of the circle. A is any point on minor arc BC. Find the value of $\angle BAC-\angle OBC$.