Theorem of Equal Angles Subtended by Different Chords

Q.

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

Q.

Let the latus rectum of the parabola $y^2=4x$ be the common chord to the circles $C_1$ and $C_2$ each of them having radius $2\surd 5$. Then, the distance between the centres of the circles $C_1$ and $C_2$ is

1. $8$

2. $8\sqrt{5}$

3. $4\sqrt{5}$

4. $12$

Q.

Find the diameter of a circle whose radius is $6cm$.

Q.

If the two chords in a circle are equal, then the angles they subtend at the centre are the same.

1. False

2. True

Q. Degree measure theorem

Q. If ABCD is a quadrilateral in which AB\parallel CD AND AD=BC .Prove that \angle A=\angle B

Q. In the figure, O is the centre of the circle. Find the length of CD, if AB=5 cm.

1. 4 cm
2. 10 cm
3. 5 cm
4. 3 cm

Q. 20. In ABC, AB = AC. If the interior circle ofABC touches the sides AB, BC and CA at D, E, F respectively. Prove that E bisects BC.

Q. 39 ABCD is a cyclic quadrilateral whose diagonals AC and BD intersect at P.If AB = DC, then prove that triangle PAB is congurence to triangle PDC

Q. 26. If a hexagon ABCDEF circumscribe a circle, prove that AB + CD + EF = BC + DE + AF

Q. If AB and CD are two equal chords of a circle with centre O such that ∠AOB = 80°, then find ∠COD.

1. 100°
2. 120°
3. 80°
4. 40°

Q. In the given figure, O is the centre of the circle. If $\angle ACB={38}^{\circ },then\angle AOBis$

1. 
   ${45}^{\circ }$
2. 
   ${19}^{\circ }$
3. 
   
   ${57}^{\circ }$
4. 
   ${76}^{\circ }$

Q. In the given figure, $\angle ABC={69}^{\circ }$ and $\angle ACB={31}^{\circ }.$ Find $\angle BDC.$

1. ${60}^{\circ }$
2. ${70}^{\circ }$
3. ${100}^{\circ }$
4. ${80}^{\circ }$

Q. Angle subtended by a diameter in a semi-circle is ____ degrees

1. 30
2. 45
3. 60
4. 90

Q. Angle ABC and Angle BCD of a cyclic quadrilateral ABCD are 75° and 105° respectively. If sum of lengths AD and AC is 100cm, then the sum of the lengths of BC and BC is

Q. O is the centre of the given circle, AB and CD are chords such that ON < OM. Which of the following is not correct?

1. 
   $AN>CM$
2. 
   $O{A}^2=O{N}^2+A{N}^2$
3. 
   $C{M}^2=O{C}^2+O{M}^2$
4. 
   $AB>CD$

Q.

Assertion: If AB = CD, then BE = DE and AE = CE, where E is the point of intersection of AD and BC.

Reason: Angles in the same segment are equal.


Which of the following options is correct?

1. Both Assertion and Reason are true and Reason is the correct explanation of Assertion.

2. Both the assertion and reason are true, but reason is not the correct explanation for assertion.

3. Assertion is true, but reason is false.

4. Assertion is false, but reason is true.

Q. in the given figure, P, Q and R are the mid-points of sides AB. AC and BC of triangle ABC respectuely IF AD is perpendicular to BC. then prove that PQRD is a cyclic quadrlateral.

Q. ABCD is a parelle\log ram. L and M are points on AB and DC respectively AL=CM . Prove that LM and BD bisect each oth

Q.

In the given figure, O is the centre of the circle. Find the value of $X$.

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

2. ${50}^{\circ }$

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

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

Q.

In the given figure, if $XY=6cm$ and $\angle XOY=\angle MON={85}^{\circ }$, then find the length of the chord MN in cm.

1. 6
2. 8
3. 4
4. 5

Q. In the given figure, what is the value of $x$?

1. 5 cm
2. 6 cm
3. 4 cm
4. 10 cm

Q.

In the given figure, if $XY=6cm$ and $\angle XOY=\angle MON={85}^{\circ }$, then find the length of the chord MN in cm.

1. 4
2. 8
3. 6
4. 5

Q. Two equal chords, $AB$ and $CD$ are at a distance of 10 cm from each other. The distance of chord $AB$ from the centre .

1. $6cm$
2. $5cm$
3. $8cm$

Q.

If the angles subtended by two chords of a circle at the centre are equal, then the lengths of the chords are unequal.

1. False

2. True

Q. In the given figure, $\angle ABC={69}^{\circ }$ and $\angle ACB={31}^{\circ }.$ Find $\angle BDC.$

Q.

In the given figure, if $XY=6cm$ and $\angle XOY=\angle MON={85}^{\circ }$, then find the length of the chord MN in cm.

1. 4
2. 8
3. 6
4. 5

Q.

If the two chords in a circle are equal, then the angles they subtend at the centre are the same.

1. True

2. False

Q. Two equal chords, AB and CD are at a distance of 10 cm from each other. Find the distance of chord AB from the centre.

Q. Two equal chords, $AB$ and $CD$ are at a distance of 10 cm from each other. The distance of chord $AB$ from the centre .

1. $6cm$
2. $5cm$
3. $8cm$