SEBA Class 9 Maths Chapter 10 Circles

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SEBA Class 9 Maths Chapter 10 Circles

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Circles

Chapter: 10

Exercise 10.1

Q.1. Fill in the blanks:

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

Ans: The centre of a circle lies in the interior of the circle.

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

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

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

Ans: The longest chord of a circle is a diameter of the circle.

(iv) An circles a_______ when its ends are the ends of a diameter.

Ans: 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 _______of the circle.

Ans: 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 ______parts.

Ans: A circle divides the plane, on which it lies, in three parts.

Q.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.

Ans: True because all points on the circle equidistant from its centre.

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

Ans: False because there are infinitely many points on the circle. So for each point on the circle, a point can be determined on the circle at a given distance from that point resulting into greatly many equal chords.

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

Ans: False because for each arc, the remaining arc will have greater length.

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

Ans: True because of definition of diameter

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

Ans: False by virtue of its definition.

(vi) A circle is a plane figure.

Ans: True as it is a part of a plane.

Exercise 10.2

Q.1. Recall that two circles are congruent if they have the same radii. Prove that equal chords of a congruent circle subtend equal angles at their centres.

Ans: Given: AB and CD are two equal chords of congruent circles with centres O and O’ respectively.

To Prove: ∠AOB= ∠CO’D

Proof: In ∆AOB and ∆ O’CD,

OA = O’C | Radii of congruent circles

OB = O’D | Radii of congruent circles

AB = CD | Given

∴ ∆ OAB ≅ O’CD | SSS Rule

∴ ∠AOB ≅ ∠ COD | c.p.c.t.

∴ ∠AOB= ∠ CO’D

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

Ans: ∠ AOB and ∠ CO’D are the two equal angles subtended by the chords AB and CD of two congruent circles with centres O and O’ respec- tively.

To Prove: AB = CD

Prove: In ΔΟΑΒ and ∆O’CD,

OA = O’C | Radii of congruent circles

OB = O’D | Radii of congruent circles

∠AOB = ∠CO’D | Given

∴ ΔΟΑΒ ≅ O’CD | SAS Rule

∴ AB = CD | c.p.c.t.

3. Which of the following statements is true or false?

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

(Q) Equal chords of a circle subtend equal angles at the centre.

(a) P is true, Q is false

(b) P is false, Q is true

(d) Both P and Q are false

Ans. (P) A circle has only finite number of equal chords -False.

[Correct Statement: A circle contains infinite number of chords.]

(Q) Equal chords of a circle subtend equal angles at the centre -True.

The correct option is (b)

Exercise 10.3

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

Ans: Each pair has at the most two common points.

(i)

No point common

(ii)

One point common

(iii)

Two points common

The maximum number of common points is Two.

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

Ans: Steps of Construction:

(i) Take any three points P, Q and R on the circle

(ii) Join PQ and QR

(iii) Draw the perpendicular bisectors PQ and QR. Let these intersect at O.

Then, O is the centre of the circle.

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

Ans: Given: Two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.

Prove: OP is the perpendicular bisector of AB.

Construction: Join OA, OB, PA and PB.

Let OP intersect AB at M.

Proof: In ∆OAP and ∆OPB,

OA=OB | Radii of a circle

PA = PB | Radii of a circle

OP=OP | Common

∴ ∆ OAP ≅ ∆ OBP | SAS Rule

∴ ∠ AOP ≅∠BOP | c.p.c.

⇒ ∠AOM ≅ ∠BOM …(1)

In ∆AOM and ∆ BOM,

OA = OB | Radii of a circle

∠AOM =∠BOM | From (1)

OM = OM | Common

∴ ∆ AOM ≅ ∆BOM | SAS Rule

∴ AM = BM …(2) | c.p.c.t.

and ∠ AMO =∠BMO …(3) | c.p.c.t.

But ∠AMO+∠BMO = 180⁰ | Linear Pair Axiom

∴ ∠AMO=∠BMO = 90⁰ …(4)

∴ OM, i.e., OP is the perpendicular bisector of AB. | From (2) and (4)

4. Match the Columns.

Column – I	Column – II
(P) No. of circles passing through three given non-collinear point is ____________	1. Perpendicular
(Q) The centre of the circle that passes through the vertices of a triangle is ___________	2. Infinite
(R) The line drawn through the centre of a circle to the chord. to bisect a chord is ____________	3. One
(S) No. of circles that can be drawn through two given points is ____________	4. Circumcentre

The correct option is –