एनसीईआरटी क्लास नाइंथ मैथ्स चैप्टर 10 सर्कल्स इंट्रोडक्शन
Answers
Answer:
Ex 10.1 Class 9 Maths 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.
Solution:
(i) interior
(ii) exterior
(iii) diameter
(iv) semicircle
(v) the chord
(vi) three
Ex 10.1 Class 9 Maths Question 2.
Write True or False. Give reason 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.
Solution:
(i) True [∵ All points on the circle are equidistant from the centre]
(ii) False [ ∵ A circle can have an infinite number of equal chords]
(iii) False [∵ Each part will be less than a semicircle]
(iv) True [ ∵ Diameter = 2 x Radius]
(v) False [ ∵ The region between the chord and its corresponding arc is a segment]
(vi) True [ ∵ A circle is drawn on a plane]
NCERT Solutions for Class 9 Maths Chapter 10 Circles (वृत्त) (Hindi Medium) Ex 10.1
NCERT Solutions for Class 9 Maths Chapter 10 Circles - वृत
NCERT Solutions for Class 9 Maths Chapter 10 Circles Exercise 10.1
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.2
Ex 10.2 Class 9 Maths 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
Solution:
Given: Two congruent circles with centres O and O’ and radii r, which have chords AB and CD respectively such that AB = CD.
To Prove: ∠AOB = ∠CO’D
Proof: In ∆AOB and ∆CO’D, we have
AB = CD [Given]
OA = O’C [Each equal to r]
OB = O’D [Each equal to r]
∴ ∆AOB ≅ ∆CO’D [By SSS congruence criteria]
⇒ ∠AOB = ∠CO’D [C.P.C.T.]
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.2 A1
Ex 10.2 Class 9 Maths Question 2.
Prove that, if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
Solution:
Given: Two congruent circles with centres O & O’ and radii r which have chords AB and CD respectively such that ∠AOB = ∠CO’D.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.2 A2
To Prove: AB = CD
Proof: In ∆AOB and ∆CO’D, we have
OA = O’C [Each equal to r]
OB = O’D [Each equal to r]
∠AOB = ∠CO’D [Given]
∴ ∆AOB ≅ ∆CO’D [By SAS congruence criteria]
Hence, AB = CD [C.P.C.T.]
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.3
Ex 10.3 Class 9 Maths Question 1.
Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?
Solution:
Let us draw different pairs of circles as shown below:
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.3 A1
We have
Figure Maximum number of common points
(i) nil
(ii) one
(«i) two
Thus, two circles can have at the most two points in common.
Ex 10.3 Class 9 Maths Question 2.
Suppose you are given a circle. Give a construction to find its centre.
Solution:
Steps of construction :
Step I : Take any three points on the given circle. Let these points be A, B and C.
Step II : Join AB and BC.
Step III : Draw the perpendicular bisector, PQ of AB.
Step IV: Draw the perpendicular bisector, RS of BC such that it intersects PQ at O.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.2 A2A
Thus, ‘O’ is the required centre of the given drcle.
Ex 10.3 Class 9 Maths Question 3.
If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.
Solution:
We have two circles with centres O and O’, intersecting at A and B.
∴ AB is the common chord of two circles and OO’ is the line segment joining their centres.
Let OO’ and AB intersect each other at M.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.3 A3
∴ To prove that OO’ is the perpendicular bisector of AB,
we join OA, OB, O’A and O’B. Now, in ∆QAO’ and ∆OBO’,
we have
OA = OB [Radii of the same circle]
O’A = O’B [Radii of the same circle]
OO’ = OO’ [Common]
∴ ∆OAO’ ≅ ∆OBO’ [By SSS congruence criteria]
⇒ ∠1 = ∠2 , [C.P.C.T.]
Now, in ∆AOM and ∆BOM, we have
OA = OB [Radii of the same circle]
OM = OM [Common]
∠1 = ∠2 [Proved above]
∴ ∆AOM = ∆BOM [By SAS congruence criteria]
⇒ ∠3 = ∠4 [C.P.C.T.]
But ∠3 + ∠4 = 180° [Linear pair]
∴∠3=∠4 = 90°
⇒ AM ⊥ OO’
Also, AM = BM [C.P.C.T.]
⇒ M is the mid-point of AB.
Thus, OO’ is the perpendicular bisector of AB.