*NCERT Solutions of Chapter: Circles. NCERT Solutions along with worksheets and notes for Class 9.*

**Exercise 10.1**

**(1)** **Fill in the blanks:**

**(i) The centre of a circle lies in ______ of the circle. (exterior/interior)**

**(ii) A point, whose distance from the centre of a circle is greater than its radius lies in ______ of the circle. (exterior/ interior)**

**(iii) The longest chord of a circle is a _____ of the circle.**

**(iv) An arc is a ____ when its ends are the ends of a diameter.**

**(v) Segment of a circle is the region between an arc and _____ of the circle.**

**(vi) A circle divides the plane, on which it lies,in _____ parts.**

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**(i)** The centre of a circle lies in **interior **of the circle. (exterior/interior)

**(ii)** A point, whose distance from the centre of a circle is greater than its radius lies in **exterior** of the circle. (exterior/ interior)

**(iii)** The longest chord of a circle is a **diameter **of the circle.

**(iv)** An arc is a **semicircle** when its ends are the ends of a diameter.

**(v)** Segment of a circle is the region between an arc and **the chord** of the circle.

**(vi)** A circle divides the plane, on which it lies, in **three **parts.

**(2)** **Write True or False: Give reasons for your answers.**

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

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

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

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

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

**(vi) A circle is a plane figure.**

**Ans-**

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

**True **

**Reason** – The distant from center of the circle to the boundary of circle is known as radius.

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

**False**

**Reason-**

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

**False**

**Reason** – If a circle is divided into three equal arcs, each will be equal arc.

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

**True**

**Reason **– diameter of the circle is twice of the radius of the circle.

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

**False**

**Reason –** The region between an arc and two radii is known as sector of a circle.

**(vi)** A circle is a plane figure.

**True**

**Reason –** A circle is a two dimensional figure.

**Exercise 10.2**

**(1) Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.**

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**(2)** **Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.**

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**Exercise 10.3**

**(1)** **Draw different parts of circles. How many points does each pair have in common? What is the maximum number of common points?**

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**Different pairs of the circles are showing below:-**

**(i)**

In figure above,

Only one point P is common. Because, both circle are passes through point P.

**(ii)**

In figure above,

Two points P and Q are common. Because, both circles are passing through points P and Q.

**(2)** **Suppose you are given a circle. Give a construction to find its centre.**

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**(3)** **If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.**

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**Exercise 10.4**

**(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.**

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**(2) If two equal chords of a circle intersect with in the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.**

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**(3) If two equal chords of a circle intersect with in the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.**

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**(4)** **If a line intersects two concentric circles (circles with the same centre ) with centre O at A, B, C and D, prove that AB = CD(Figure below).**

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**(5)** **Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6 m each, what is the distance between Reshma and Mandip?**

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**(6)** **A circular park of radius 20m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the sting of each phone.**

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**(2)** **A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at appoint on the major arc.**

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**Exercise 10.6**

**(1)** **Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.**

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**(2)Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6 cm, find the radius of the circle.**

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**(3)** **The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the centre, what is the distance of the other chord from the centre?**

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**(4)** **Let the vertex of an angle ABC be located outside a circle and let the sides of the angle intersect equal chords AD and CE with the circle. Prove that ABC is equal to half the difference of the angles subtended by the chords AC and DE at the centre.**

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**(5)** **Prove that the circle drawn with any side of a rhombus as diameter, passes through the point of intersection of its diagonals.**

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**(6) ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Prove that AE = AD.**

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**Helping Topics**