Theorem 1: Equal Chords Subtend Equal Angles at the Center

Q. In the given pentagon ABCDE, AB = BC = CD = DE = AE. What is the value of x?

1. ${36}^{\circ }$
2. ${54}^{\circ }$
3. ${108}^{\circ }$
4. ${72}^{\circ }$

Q.

A circle touches the $y$-axis at the point $(0,4)$and cuts the $x$-axis in a chord of length $6$ units. The radius of the circle is

1. $3$

2. $4$

3. $5$

4. $6$

Q.

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

___.

Q. AB = 36 cm and CD = 48 cm are two parallel chords of a circle with centre O. The distance between them is 42 cm. The radius of the circle is:

1. 27cm
2. 25 cm
3. 30 cm
4. 33 cm

Q.

In the figure, $O$ is the centre of the circle of radius $5\text{cm}.$ $P$ and $Q$ are points on chords $AB$ and $CD$ respectively such that $OP\perp AB,OQ\perp CD,AB||CD,AB=6\text{cm}$ and $CD=8cm.$ Determine $PQ$.

1. $5cm$

2. $6cm$

3. $7cm$

4. $8cm$

Q.

If (x+y)^-1(x^-1+y^-1)= x^p y^q, then p+q is equal to

1. 2

2. -1

3. -2

4. 1

Q.

Question 1
Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.

Q. A chord of length 14 cm is at a distance of 6 cm from the centre of a circle. The length of another chord at a distance of 2 cm from the centre of the circle is

Q.

In the given figure, $△ABC$ is an isosceles triangle with $AB=AC$ and $O$ is the centre of the circle. If $\angle AOB=\alpha$ and $\angle AOC=\beta$, then which of the following is true?

1. $\alpha =2\beta$

2. $\alpha =\frac{\beta }{2}$

3. $\alpha =\beta$

4. $\alpha =3\beta$

Q.

AB and CD are the chords of a circle whose center is O. They intersect each other at P. If PO is the bisector of $\angle APD$, prove that AB = CD. [2 MARKS]

Q.

P and Q are points on intersection of two circles with centre O and O'. If straight line APB and CQD are parallel to OO' then AB = ___

Q. Angle subtended by one of the equal chords at centre is 60 degrees, angle made by other chord is:

1. 30 degrees
2. 60 degrees
3. 12 degrees
4. 45 degrees

Q.

If a line intersects two concentric circles (circles with the same centre) with centre $\mathrm{O}$ at $\mathrm{A},\mathrm{B},\mathrm{C}$ and $\mathrm{D}$, prove that $\mathrm{AB}=\mathrm{CD}$ (see Fig) .

Q.

Question 5
In the given figure, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that $\angle BEC={130}^{\circ }$ and $\angle ECD={20}^{\circ }$. Find $\angle BAC$.

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

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

Q. Question 2
In figure, if OA = 5 cm, AB = 8 cm and OD is perpendicular to AB, then CD is equal to


(A) 2 cm
(B) 3 cm
(C) 4 cm
(D) 5 cm

Q.

Two chords AB, CD of length 5 cm, 11 cm respectively of a circle are parallel. If the distance between AB and CD is 3 cm. find the radius of the circle.Assume that the chords are on the same side of the center.

1. 5.02cm

2. 7.02cm

3. 8.02cm

4. 6.02cm

Q.

Two chords $AB$ and $AC$ of a circle subtends angles equal to ${90}^{\circ }$ and ${150}^{\circ }$ respectively at the centre. Find $\angle BAC$.

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

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

3. ${80}^{\circ }$

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

Q.

Two chords AB and AC of a circle subtends angles equal to ${90}^{\circ }and{150}^{\circ }$ respectively at the centre. Find $\angle BAC$, if AB and AC lie on the opposite sides of the centre.

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

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

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

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

Q. Question 9
Two chords AB and AC of a circle subtends angles equal to ${90}^{\circ }and{150}^{\circ }$ respectively at the centre. Find $\angle BAC$, if AB and AC lie on the opposite sides of the centre.

Q.

In the given figure, lengths of the chords $AB$ and $CD$ are $12cm$ and $18cm$ respectively and distance between them is $15cm$. Find the radius of the circle.

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. 35

3. 45

4. 55

Q.

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

1. 55

2. 35
3. 45
4. 25

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.

Two arcs APB and CQD of a circle are in the ratio 5 :7. The angle subtended by arc APB at the centre is ${150}^o$.

1. 1

2. 2

Q. 35. Two circles A and B of radius 3 cm and 4cm intersect at the point C and D such thatcAC and BC are tangent to the circle. Find the lenght of common chord CD

Q.

Given one line and one circle that you can arrange in any way you like, the minimum number of points where they intersect is

1. 0

2. 1

3. 2

4. Infinity

Q.

A chord 24cm long is drawn in a circle of radius 13cm. Find its distance from the centre.

1. 5 cm

2. 50 cm

3. 4 cm

4. 14 cm

Q. Which of the following is/are true?

1. In a circle, if two chords are equal in measure, then their corresponding minor arcs are equal in measure.

2. If two arcs of a circle are equal, the subtend equal angles at the centre.
3. Circles having different radii are similar.
4. Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.

Q. O is the centre of the circle. $\angle OAB={20}^{\circ },\angle OCB={55}^{\circ }$. Find $\angle BOC$ and $\angle AOC$.