• Skip to main content
  • Skip to primary sidebar
  • Skip to footer
  • NCERT Solutions
    • NCERT Books Free Download
  • TS Grewal
    • TS Grewal Class 12 Accountancy Solutions
    • TS Grewal Class 11 Accountancy Solutions
  • CBSE Sample Papers
  • NCERT Exemplar Problems
  • English Grammar
  • MCQ Questions

CBSE Tuts

CBSE Maths notes, CBSE physics notes, CBSE chemistry notes

NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5

NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5 are part of NCERT Solutions for Class 9 Maths. Here we have given NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.

  • 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 10.5
Number of Questions Solved 11
Category NCERT Solutions

NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5

Ex 10.5 Class 9 Maths Question 1.
In the figure, A, B, and C are three points on a circle with center O such that ∠BOC = 30° and ∠AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ADC.
Solution.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.1
Here, ∠AOC = ∠AOB + ∠BOC = 60° + 30° = 90°
Since, arc ABC makes an angle of 90° at the center  the circle.
∴ ∠ADC =  \(\cfrac { 1 }{ 2 } \) ∠AOC
[since, the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle]
= \(\cfrac { 1 }{ 2 } \) x 90° = 45°

Ex 10.5 Class 9 Maths 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.
Solution.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.2
Let PQ be chord. Join OP and OQ.
It is given that PQ = OP = OQ (∵  Chord = radius
Δ OPQ is equilateral.
⇒ ∠POQ = 60°
Since arc PBQ makes reflex ∠POQ = 360 °-60 ° = 300 ° at the center of the circle and ∠PBQ at a point in the minor arc of the circle.
∠PBC = \(\cfrac { 1 }{ 2 } \) (reflex ∠POQ)
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.3

Ex 10.5 Class 9 Maths Question 3.
In the figure, ∠PQR = 100°, where P, Q, and R are points on a circle with center O. Find ∠OPR.
Solution.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.4

Ex 10.5 Class 9 Maths Question 4.
In figure, ∠ABC = 69° and ∠ACB = 31°. Find ∠BDC.
Solution.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.5

Ex 10.5 Class 9 Maths Question 5.
In the figure, 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.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.6
Solution.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.7
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.8

Ex 10.5 Class 9 Maths 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 q ∠ECD.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.9
Solution.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.10

Ex 10.5 Class 9 Maths Question 7.
If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, then prove that it is a rectangle.
Solution.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.11
Given: Diagonals NP and QM of a cyclic
quadrilateral NQPM are diameters of the circle passing through the vertices M, P, Q, and N.
To prove : Quadrilateral NQPM is a rectangle.
Here, ON = OP = OQ = OM [radii of same circle]
Then,  ON = OP = \(\cfrac { 1 }{ 2 } \) NP
and  OM = OQ = \(\cfrac { 1 }{ 2 } \) MQ
∴  NP = MQ
Hence, the diagonals of the quadrilateral NQPM are equal and bisect each other. So, quadrilateral NQPM is a rectangle.
Hence proved

Ex 10.5 Class 9 Maths Question 8.
If the non-parallel sides of a trapezium are equal, then prove that it is cyclic.
Solution.
Given: Non-parallel sides PS and QR of a trapezium PQRS are equal. To prove: PQRS is a cyclic trapezium.
Construction: Draw SM ⊥  PQ and RN ⊥  PQ.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.12

Ex 10.5 Class 9 Maths 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, P and Q respectively (see figure). Prove that ∠ACP = ∠QCD.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.13
Solution.
Given: Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn which intersect the circles at A, D, P and Q, respectively
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.14

Ex 10.5 Class 9 Maths Question 10.
If circles are drawn taking two sides of a triangle as diameters, then prove that the point of intersection of these circles lies on the third side.
Solution.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.15
Given: Two circles are drawn with sides AC and
AB of Δ ABC as diameters. Both circles intersect each other at D.
To prove : D lies on BC.
Construction: Join AD.
Proof : Since, AC and AB are the diameters of the two circles.
∴  ∠ADB = 90°     …(i) [angle in semi-circle]
and  ∠ADC = 90°  …(ii) [angle in semi-circle]
On adding Eqs. (i) and (ii), we get
∠ADB + ∠ADC = 90°+ 90° = 180°
∠BDC = 180°
Hence, BDC is a straight line.
So, point of intersection D lies on the third side.
Hence proved.

Ex 10.5 Class 9 Maths Question 11.
ABC and ADC are two right angled triangles with common hypotenuse AC. Prove that ∠CAD = ∠CBD.
Solution.
Since Δ ADC and Δ ABC are right angled triangles with common hypotenuse.
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.16
Draw a circle with AC as diameter passing through B and D. Join BD.
∵ Angles in the same segment are equal.
∴ ∠CBD = ∠CA

Ex 10.5 Class 9 Maths Question 12.
Prove that a cyclic parallelogram is a rectangle.
Solution.
Given : PQRS is a parallelogram inscribed in a circle.
To prove: PQRS is a rectangle.
Proof : Since, PQRS is a cyclic quadrilateral.
∠P + ∠R = 180°   …(i)
[since, sum of pair of opposite angles in a cyclic quadrilateral is 180°]
∠P=∠R  …(ii) [since, in a parallelogram, opposite angles are equal]
NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5.17

 

We hope the NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5 help you. If you have any query regarding NCERT Solutions for Class 9 Maths Chapter 10 Circles Ex 10.5, drop a comment below and we will get back to you at the earliest.

Primary Sidebar

NCERT Exemplar problems With Solutions CBSE Previous Year Questions with Solutoins CBSE Sample Papers

Recent Posts

  • MCQ Questions with Answers for Class 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, and 1 all Subjects
  • Angle, Types of Angles
  • AND gate is formed by using two? 1)OR 2)NAND 3)NOT 4)NOR
  • And expression for one minus the quotient of one and a number X?
  • What is average human body temperature in Kelvins?
  • How many moles of caffeine are in a cup, and how many molecules of caffeine?
  • How far will the car have traveled in that time?
  • What is its atomic number?
  • How many neutrons does it have?
  • An atom loses electrons to form what?
  • What is the atomic number of this atom?
  • Which one of these is the answer?
  • What is its concentration?
  • Can an equilateral triangle also be isosceles?
  • What is the charge of an alpha particle?

Footer

Maths NCERT Solutions

NCERT Solutions for Class 12 Maths
NCERT Solutions for Class 11 Maths
NCERT Solutions for Class 10 Maths
NCERT Solutions for Class 9 Maths
NCERT Solutions for Class 8 Maths
NCERT Solutions for Class 7 Maths
NCERT Solutions for Class 6 Maths

SCIENCE NCERT SOLUTIONS

NCERT Solutions for Class 12 Physics
NCERT Solutions for Class 12 Chemistry
NCERT Solutions for Class 11 Physics
NCERT Solutions for Class 11 Chemistry
NCERT Solutions for Class 10 Science
NCERT Solutions for Class 9 Science
NCERT Solutions for Class 7 Science
MCQ Questions NCERT Solutions
CBSE Sample Papers
cbse ncert
NCERT Exemplar Solutions LCM and GCF Calculator
TS Grewal Accountancy Class 12 Solutions
TS Grewal Accountancy Class 11 Solutions