Theorem of Equal Chords Subtending Equal Angles at the Center

Q.

If $O$ is the point of intersection of two chords $AB$ and $CD$ of a circle such that $OB=OD$, then prove that triangles $△OAC$ and $△ODB$ are similar.

Q.

In the given figure, chords $AB=CD,\angle COD={60}^{\circ }$ and $\angle AOD={170}^{\circ }$.

Find the value of $\angle BOC$.

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

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

3. ${75}^{\circ }$

4. ${80}^{\circ }$

Q.

In the given figure, AB and CD are two equal chords of a circle. The angle ODC will be equal to

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

2. ${55}^{\circ }$

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

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

Q.

In the given figure, it is given that chords AB = PQ. The $\angle AOB={55}^0$, and $\angle POQ=x^0$, find $x$.

1. 55

2. 25
3. 45
4. 35

Q.

ABCD is a cyclic quadrilateral in a circle with centre O. If $angleADC={130}^o$ Find $\angle BAC$

1. ${120}^o$

2. ${80}^o$

3. ${40}^o$

4. ${134}^o$

Q.

AB is a chord to a circle with Centre O. X and Y are points on the circumference of the circle. Lines are drawn connecting A and B with O, X and Y. Match the following on the basis of angles made by A and B.

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 - (c)
   III- (a)

Q.

ABCD is a cyclic quadrilateral whose side AB is a diameter of the circle. If $\angle ADC={130}^{\circ }$, find $\angle BAC$.

1. $(\frac{120}{2}{)}^{\circ }$

2. $(\frac{120}{3}{)}^{\circ }$

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

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

Q.

In the figure, it is given that XY = 6 cm,
$\angle XOY=\angle MON={85}^{\circ }$.
The length of the chord MN is ____ cm.

1. 4

2. 8

3. 6

4. 5

Q. Question 3
If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.

Q. Consider the below figure. Here, $z$ = ___.

1. ${45}^{\circ }$
2. ${15}^{\circ }$
3. ${60}^{\circ }$
4. ${30}^{\circ }$

Q.

Equal chords subtend equal angles at the centre.

1. True
2. False

Q.

If AB is a chord to a circle with Centre O. X and Y are points on the circumference of the circle. Lines are drawn connecting A and B with O, X and Y. Match the following on the basis of angles made by A and B.

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 - (c)
   III- (a)

Q.

In the figure below, it is given that chords AB = PQ. The $\angle AOB={55}^0$, and $\angle POQ=x^0$, find $x$.

1. 25

2. 55

3. 35
4. 45

Q.

In the following figure, what can be concluded from the below given conditions.

Given: AB = CD and AC = DE.

1. All of the above.

2. AC = BD, $\angle$AOC = $\angle$BOD

3. $\angle$AOC = $\angle$DOE, AC = DE

4. AD = CE, $\angle$AOD = $\angle$COE.

Q.

The outer wheel of a ferris wheel is made of several steel arcs. If two arcs of equal length AB and PQ are taken, and the chord forming arc AB is measured to be 60 cm. What would be the length of chord forming arc PQ?

1. 4 cm
2. 3 cm
3. 6 cm
4. 12 cm

Q.

AB and PQ are two equal chords of a circle. If the length of minor arc and major arc formed by AB is 24cm and 64cm respectively, then the length of minor and major arc formed by PQ is ___ and ___ respectively.

1. 24 cm, 64 cm

2. 64 cm, 24 cm

3. 32 cm, 12 cm

4. 12 cm, 32 cm

Q.

In the given figure, it is given that chords $AB=PQ,\angle AOB={55}^0,$ and $\angle POQ=x^0$, find $x$.

1. 25

2. 45

3. 55

4. 35

Q.

O is the centre of the circle. AB is a chord of the circle. Angle subtended by the chord at the center and the angle subtended by minor arc AB are equal. Is it true or false?

1. True

2. False

Q. If the angles subtented by a minor arc is 30 degree, then what is the angle subtended by the major arc?
$\left[2Marks\right]$

Q.

If two arcs of a circle are equal, then length of their corresponding chords will be

___.

Q.

In fig., circles C(O, r) and $C(O{}^{\prime },\frac{r}{2})$ touch internally at a point A and AB is a chord of the circle C (O, r) intersecting $C(O{}^{\prime },\frac{r}{2})$ at C. Prove that AC = CB. [3 MARKS]