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NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4

NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4 are part of NCERT Solutions for Class 9 Maths. Here we have given NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4.

  • Circles Class 9 Ex 10.1
  • Circles Class 9 Ex 10.2
  • Circles Class 9 Ex 10.3
  • Circles Class 9 Ex 10.4
  • Circles Class 9 Ex 10.5
  • Circles Class 9 Ex 10.6
Board CBSE
Textbook NCERT
Class Class 9
Subject Maths
Chapter Chapter 10
Chapter Name  Quadrilaterals
Exercise  Ex 10.4
Number of Questions Solved 6
Category NCERT Solutions

NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4

Ex 10.4 Class 9 Maths Question 1.
Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centers is 4 cm. Find the length of the common chord.
Solution.
Let O and O’ be the centers of the circles of radii 5 cm and 3 cm, respectively. Let AB be their common chord
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4.1
Given, OA = 5 cm, O’ A = 3 cm and 00′ = 4 cm
Then, AO’2 + OO’2 = 32 + 42 = 9 +16 = 25 = OA2
So, 00′ A is a right angled triangles and right angled at O’.
∴ Area of Δ OO’ A = \(\cfrac { 1 }{ 2 } \) x O’A x OO’ = \(\cfrac { 1 }{ 2 } \) x 3 x 4 = 6 cm2 …(i)
Also, we know that when two circles intersect at two points, then their centers lie on the perpendicular bisector of the common chord.
∴ Area of Δ OO’ A = \(\cfrac { 1 }{ 2 } \) x OO’ x AM = \(\cfrac { 1 }{ 2 } \) x 4 x AM=2AM ………(ii)
From eqs. (i) and (ii), we get
2AM = 6 ⇒ AM =3 cm
∴ AB = 2 x AM = 2 x 3 = 6 cm

Ex 10.4 Class 9 Maths 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.
Solution.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4.2
Let MN and AB are two chords of a circle with center 0. AB and MN intersect at P and MN = AB
To prove : MP = PB and PN = AP
Construction: Draw OD  ⊥ L MN and OC ⊥ AB. Join OP.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4.3

Ex 10.4 Class 9 Maths Question 3.
If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the center makes equal angles with the chords.
Solution.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4.4
Let RQ and MN be two equal chords of a circle with center O.
MN and RQ intersect at P and MN = RQ.
To prove : ∠OPM = ∠OPQ
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4.5

Ex 10.4 Class 9 Maths Question 4.
If a line intersects two concentric circles (circles with the same center) with center O at A, B, C and D, then prove that AB = CD (See figure).
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4.6
Solution.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4.7
Let a line l intersects two concentric circles with center O at A, B, C and D.
To prove  : AB   = CD
Construction: Draw OP perpendicular from O on line l.
Proof: We know that the perpendicular drawn from the center of a circle to a chord bisects the chord. Here, BC is the chord of the smaller circle and OP ⊥ BC.
∴  BP  = PC
and AD is a chord of the larger circle and OP ⊥ AD.
∴   AP    = PD              …(ii)
On subtracting eq. (i) from eq. (ii), we get
AP -BP = PD -PC
⇒ AB = CD
Hence proved.

Ex 10.4 Class 9 Maths Question 5.
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?
Solution.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4.8
Let 0 be the center of the circle and Reshma,
Salma and Mandip are represented by the points R, S, and M, respectively.
Draw OP ⊥ RM and ON ⊥ RS.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4.9
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4.10

Ex 10.4 Class 9 Maths Question 6.
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.
Solution.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4.11
Let Ankur, Syed and David be sitting on the points P, Q, and R, respectively on the boundary of circular park.
Clearly, PQ = QR = PR, as they sitting at an equal distance.
Thus, ΔPQR is an equilateral triangle.
Let  PQ = QR = PR = x m
Now, draw altitudes PC, QD and RN from vertices to .the sides of a triangle and these altitudes intersect at the center of circle M.
[∵  altitudes of equilateral triangle passes through the circumcenter of the equilateral triangle.]
As Δ PQR is an equilateral triangle, therefore these altitudes bisect their sides.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4.12
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4.13

We hope the NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4 help you. If you have any query regarding NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.4, drop a comment below and we will get back to you at the earliest.

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