A Line through the Center That Bisects the Chord Is Perpendicular to the Chord.

Q.

Two circles of radii $5cm$ and $3cm$ intersect at two points and the distance between their centers is $4cm.$ The length of the common chord is

1. $10cm$

2. $5cm$

3. $6cm$

4. $8cm$

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

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

Q. Question 3
If AB = 12 cm, BC = 16 cm and AB is perpendicular to BC, then the radius of the circle passing through the points A, B and C is

(A) 6 cm
(B) 8 cm
(C) 10 cm
(D) 12 cm

Q.

A chord of length 30 cm is drawn at a distance of 8 cm from the centre of a circle. Find out the radius of the circle.

Q. Diameter of a circle is 26 cm and length of a chord of the circle is 24 cm. Find the distance of the chord from the centre.

Q.

Two circles of radii $10cm$ and $8cm$ intersect each other, and the length of the common chord is $12cm.$ The distance between their centers will be

1. $3+\sqrt{2}cm$

2. $8+2\sqrt{7}cm$

3. $8cm$

4. $2\sqrt{6}cm$

Q.

A chord is at a distance of 8 cm from the centre of a circle of radius 17 cm. The length of the chord is

(a) 25 cm

(b) 12.5 cm

(c) 30 cm

(d) 9 cm

Q.

$A,B$ and $C$ are three points on a circle such that the angles subtended by the chords $AB$ and $AC$ at the centre $O$ are ${90}^{\circ }$ and ${110}^{\circ }$ respectively. Then the measure of angle $BAC$ is

1. ${75}^{\circ }$

2. ${40}^{\circ }$

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

4. ${50}^{\circ }$

Q.

If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

Q.

If two equal chords of a circle intersect within a circle. Prove that the segment of one chord are equal to the corresponding segment of another.

Q. The line drawn through the centre of a circle to bisect a chord is _________ the chord.

1. equal to
2. perpendicular to
3. double
4. parallel to

Q.

Two equal chords $AB$ and $CD$ of a circle when produced intersect at a point $P.$ Prove that $PB=PD.$

Q. Distance of chord AB from the centre of a circle is 8 cm. Length of the chord AB is 12 cm. Find the diameter of the circle.

Q.

The radius of a circle which has a 6 cm long chord, 4 cm away from the centre of the circle is

1. 5 cm

2. 8 cm

3. 9 cm

4. 10 cm

Q.

Two chords AB and CD of a circle intersect each other at P outside the circle. If AB = 6 cm, BP = 2 cm and PD = 2.5 cm, find CD.

Q.

The length of chord which is at a distance of 12 cm from centre of circle of radius 13 cm is:

1. 12 cm

2. 5 cm

3. 13 cm

4. 10 cm

Q.

Find the length of a chord which is at a distance of 3 cm from the centre of a circle of radius 5 cm.

Q. Question 12
AB and AC are two chords of a circle of radius r such that AB = 2AC. If p an q are the distances of AB an AC from the centre, prove that $4q^2=p^2+3r^2$.

Q. In the given figure, C is the centre of the circle. seg QT is a diameter CT = 13, CP = 5, find the length of chord RS.

Q.

If two equal chords of a circle intersect, prove that the parts of one chord are separately equal to the parts of the other chord.

Q. Question 14
In the figure, O is the centre of the circle BD = OD and CD $\perp$ AB. Find $\angle CAB$.

Q. Question 7
If ABC is an equilateral triangle inscribed in a circle and P be any point on the minor arc BC which does not coincide with B or C, then prove that PA is angle bisector of $\angle BPC$.

Q. Radius of a circle is 34 cm and the distance of the chord from the centre is 30 cm, find the length of the chord.

Q.

The perpendicular distance of a chord 8 cm long from the centre of a circle of radius 5 cm is

1. 4 cm

2. 9 cm

3. 3 cm

4. 2 cm

Q. Write the truth value (T/F) fo the following with suitable reasons:

(i) A circle is a plane figure.

(ii) Line segment joining the centre to any point on the circle is a radius of the circle.

(iii) If a circle is divided into three equal arcs each is a major arc.

(iv) A circle has only finite number of equal chords.

(v) A chord of a circle, which is twice as long is its radius is a diameter of the circle.

(vi) Sector is the region between the chord and its corresponding arc.

(vii) The degree measure of an arc is the complement of the central angle containing the arc.

(viii) The degree measure of a semi-circle is 180°

Q.

Sides of a triangle ABC are equidistant from the centre O of its circumcircle. Triangle ABC is

1. Equilateral triangle

2. Scalene triangle

3. Right triangle

4. Isosceles triangle

Q. Question 3
A, B and C are three points on a circle, prove that perpendicular bisectors of AB, BC and CA are concurrent.

Q. Question 5
In the figure, if AOB is a diameter of the circle and AC = BC, then $\angle CAB$ is equal to

(A) ${30}^{\circ }$
(B) ${60}^{\circ }$
(C) ${90}^{\circ }$
(D) ${45}^{\circ }$

Q. Question 13
In the figure, O is the centre of the circle $\angle BCO={30}^{\circ }$. Find x and y.

Q. A chord is at a distance of 8 cm from the centre of a circle of radius 17 cm. The length of the chord is
(a) 25 cm
(b) 12.5 cm
(c) 30 cm
(d) 9 cm