# NCERT Solutions for Class 9 Maths Unit 10

## Circles Class 9

### Unit 10 Circles Exercise 10.1, 10.2, 10.3, 10.4, 10.5, 10.6 Solutions

Introduction to Circles
A circle is the set of all points equidistant from a given point. The point from which all the points on a circle are equidistant is called the center of the circle, and the distance from that point to the circle is called the radius of the circle. A circle is named with a single letter, its center.
All circles have a diameter, too. The diameter of a circle is the segment that contains the center and whose endpoints are both on the circle. The length of the diameter is twice that of the radius. Therefore, all diameters of a circle are congruent, too.

Keep in mind that an infinite number of radii and diameters can be drawn in a circle. Although they are all congruent, they are not the same. Sometimes a strategically placed radius will help make a problem much clearer. Likewise, diameters can be drawn into a circle to strategically divide the area within the circle. Each of these techniques is prevalent in geometric proofs, and each is based on the facts that all radii are congruent, and all diameters are congruent. However, their position when drawn makes each one different.

### Exercise 10.1 : Solutions of Questions on Page Number : 171

Q1 :

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.

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

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

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

(iv) An arc is a semi-circle when its ends are the ends of a diameter.

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

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

Q2 :

(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.

(i) True. All the points on the circle are at equal distances from the centre of the circle, and this equal distance is called as radius of the circle.

(ii) False. There are infinite points on a circle. Therefore, we can draw infinite number of chords of given length. Hence, a circle has infinite number of equal chords.

(iii) False. Consider three arcs of same length as AB, BC, and CA. It can be observed that for minor arc BDC, CAB is a major arc. Therefore, AB, BC, and CA are minor arcs of the circle.

(iv) True. Let AB be a chord which is twice as long as its radius. It can be observed that in this situation, our chord will be passing through the centre of the circle. Therefore, it will be the diameter of the circle.

(v) False. Sector is the region between an arc and two radii joining the centre to the end points of the arc. For example, in the given figure, OAB is the sector of the circle.

(vi) True. A circle is a two-dimensional figure and it can also be referred to as a plane figure.

### Exercise 10.2 : Solutions of Questions on Page Number : 173

Q1 :

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.

Q2 :

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

### Exercise 10.3 : Solutions of Questions on Page Number : 176

Q1 :

Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

Q2 :

Suppose you are given a circle. Give a construction to find its centre.

Q3 :

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

### Exercise 10.4 : Solutions of Questions on Page Number : 179

Q1 :   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.

Q2 :

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.

Q3 :

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.

Q4 :

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 (see figure 10.25).

Q5 :

Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5 m 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?

Q6 :

A circular park of radius 20 m 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 string of each phone.

### Exercise 10.5 : Solutions of Questions on Page Number : 184

Q1 :

In the given figure, A, B and C are three points on a circle with centre O such that ∠ BOC = 30° and ∠ AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ ADC.

Q2 :

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 a point on the major arc.

Q3 :

In the given figure, ∠ PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠ OPR.

Q4 :

In the given figure, ∠ ABC = 69°, ∠ ACB = 31°, find ∠ BDC.

Q5 :

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 ∠ BEC = 130° and ∠ ECD = 20°. Find ∠ BAC.

Q6 :

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠ DBC = 70°, ∠ BAC is 30°, find ∠ BCD. Further, if AB = BC, find ∠ ECD.

Q7 :

If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

Q8 :

If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

Q9 :

Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see the given figure). Prove that ∠ ACP = ∠ QCD.

Q10 :

If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

Q11 :

ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠ CAD = ∠ CBD.

Q12 :

Prove that a cyclic parallelogram is a rectangle.

### Exercise 10.6 : Solutions of Questions on Page Number : 186

Q1 :

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

Q2 :

Two chords AB and CD of lengths 5 cm 11cm 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.

Q3 :

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?

Q4 :

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.

Q5 :

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

Q6 :

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

Q7 :

AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are diameters; (ii) ABCD is a rectangle.

Q8 :

Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of the triangle DEF are 90° .

Q9 :

Two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ.

Q10 :

In any triangle ABC, if the angle bisector of ∠ A and perpendicular bisector of BC intersect, prove that they intersect on the circum circle of the triangle ABC.