NCERT Solutions for Class 9th Mathematics

Chapter 10 – CIRCLES

(Complete Downloadable Chapter Solution PDF file is at the bottom of the page)

 

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.

 

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.

 

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.

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

 

EXERCISE 10.3

 

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

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

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

 

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.

2.  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.

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.

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 (see Fig. 10.25).

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 6m each, what is the distance between Reshma and Mandip?

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 string of each phone.

 

EXERCISE 10.5

 

1.  In Fig. 10.36, 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.

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 a point on the major arc.

3.  In Fig. 10.37, ∠ PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠ OPR.

4.  In Fig. 10.38, ∠ ABC = 69°, ∠ ACB = 31°, find ∠ BDC.

5.  In Fig. 10.39, 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.

6.  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.

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

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

9.  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 Fig. 10.40). Prove that ∠ ACP = ∠ QCD.

10.  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.

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

12.  Prove that a cyclic parallelogram is a rectangle.