NCERT Solutions Class 9 Mathematics Chapter 10 Circles Download In Pdf

Chapter 10 Circles Download in Pdf

EXERCISE 10.1

**Question 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.

** Question 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

** Question 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.

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

EXERCISE 10.3

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

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

**Question 3.** If two circles intersect at two points, prove that their centres lie on the perpendicular

EXERCISE 10.4

** 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.

** Question 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.

** Question 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.

** Question 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).

** Question 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?

** Question 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

** Question 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

**Question 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.

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

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

**Question 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.

**Question 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.

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

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

**Question 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.

** Question 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.

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

**Question 12.** Prove that a cyclic parallelogram is a rectangle.

EXERCISE 10.6 (Optional)*

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

** Question 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.

** Question 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?

**Question 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.

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

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

** Question 7.** 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.

**Question 8.** 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° –
1
2
A, 90° –
1
2
B and
90° –
1
2
C.

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- Chapter 3 Coordinate Geometry
- Chapter 4 Linear Equations In Two Variables
- Chapter 7 Triangles
- Chapter 8 Quadrilaterals
- Chapter 9 Areas of Parallelograms and Triangles
- Chapter 10 Circles
- Chapter 11 Constructions
- Chapter 12 Heron’s Formula
- Chapter 13 Surface Areas and Volumes
- Chapter 14 Statistics
- Chapter 15 Probability

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