Angle Subtended by a Chord at a Point on the Circle

Q. An equilateral triangle ABC is inscribed in a circle with centre O. Find $\angle BOC$.

Q.

$AOCB$ is a quadrilateral in the circle with centre $O$. If $\angle AOC=130°$, then find $\angle ABC$.

Q. Find the value of x in the given figure if$\angle AOB=70°$ and AB = CD.

1. 50°
2. 60°
3. 70°
4. 80°

Q. 𝐀𝐁 and 𝐂𝐃 are two chords of a circle centered at 𝐎 such that 𝐀𝐁 < 𝐂𝐃. Then, which of the following is true?

1. ∠AOB > ∠COD

2. ∠AOB < ∠COD

3. ∠AOB = ∠COD

Q. Find the value of $x$ in degrees from the given figure if $\angle AOB={70}^{\circ }$ and $AB=CD.$

1. 70

Q.

Match the two columns;

$\begin{array}{cc}\text{Column-I}& \text{Column-II}\\ \text{(A) C is a point on the minor arc AB of the circle with centre O.}& {\text{(P) 60}}^{\circ }\\ \text{Given}\angle \text{ACB = x. Calculate}\angle \text{x if ACBO is a parallelogram.}& \\ \text{(B) Chord ED is parallel to the diameter AC of the circle.}& {\text{(q) 120}}^{\circ }\\ \text{Given}\angle {\text{CBE = 55}}^{\circ }\text{, calculate}\angle \text{DEC.}& \end{array}$

$\begin{array}{cc}\text{(C) In the given figure, O is the centre of the circle.}& {\text{(r) 35}}^{\circ }\\ \text{If}\angle {\text{ACB = 60}}^{\circ }\text{find}\angle \text{OAB.}& \end{array}$



$\begin{array}{cc}\text{(D) In the given figure, O is the centre of a circle.}& {\text{(s) 30}}^{\circ }\\ \angle \text{AOB = 400 and}\angle \text{BDC = 1000. Find}\angle \text{OBC}& .\end{array}$

Choose the Correct option;

1. A-s; B-p; C-q; D-r

2. A-r; B-s; C-p; D-q

3. A-q; B-p; C-r; D-s

4. A-q; B-r; C-s; D-p

Q.

BC is a chord with centre O. A is a point on an arc BC as shown in the figure. Prove that:

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(i) $\angle BAC+\angle OBC={90}^{\circ }$, if A is the point on the major arc

(ii) $\angle BAC-\angle OBC={90}^{\circ }$, if A is the point on the minor arc.
[4 MARKS]

Q. Angles subtended by equal chords at the centre of the circle are .

1. supplementary
2. equal
3. complementary

Q.

Using the information given in the figure, match the following.

1. I - (a), II - (c), III- (b)
2. I - (c), II - (a), III- (b)
3. I - (a), II - (b), III- (c)
4. I - (b), II - (a), III- (c)

Q. Perpendicular from the centre bisects the chord.

1. False
2. True

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. 120${}^{\circ }$
4. 90${}^{\circ }$

Q.

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If $\angle DBC={80}^{\circ }$ and $\angle BAC={40}^{\circ },$ then find $\angle BCD$.

1. ${60}^{\circ }$

2. ${80}^{\circ }$

3. ${50}^{\circ }$

4. ${40}^{\circ }$

Q. In the given figure, what is the angle subtended by chord AB at the point C on the circle?

1. ${60}^{\circ }$
2. ${65}^{\circ }$
3. ${55}^{\circ }$
4. ${70}^{\circ }$

Q. Find the value of $x$ in degrees from the given figure if $\angle AOB={70}^{\circ }$ and $AB=CD.$

1. 70

Q.

In the given figure, AB is the diameter of the circle. Find the value of $x$.

1. ${30}^{\circ }$

2. ${45}^{\circ }$

3. ${60}^{\circ }$

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

Q. Find the value of $x$ in the given figure if $\angle AOB={70}^{\circ }$ and $AB=CD.$

1. ${60}^{\circ }$
2. ${80}^{\circ }$
3. ${40}^{\circ }$
4. ${70}^{\circ }$

Q.

In the given figure, AB is the diameter of the circle. Find the value of $x$.

1. ${60}^{\circ }$

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

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

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

Q.

A chord AB of a circle is equal to the radius of the circle. Find the angles subtended by the chord at points on the major arc and the minor arc.

1. ${150}^{\circ },{30}^{\circ }$

2. ${120}^{\circ },{60}^{\circ }$

3. ${60}^{\circ },{120}^{\circ }$

4. ${30}^{\circ },{150}^{\circ }$