NCERT Solutions Class 9 Maths Exercise 8.2 of chapter 8-Quadrilateral - Future Study Point

NCERT Solutions Class 9 Maths Exercise 8.2 of chapter 8-Quadrilateral

Class 9 maths ncert ex 8.2

NCERT Solutions Class 9 Maths Exercise 8.2 of chapter 8- Quadrilateral

Class 9 maths ncert ex 8.2

NCERT Solutions Class 9 Maths Exercise 8.2 of chapter 8-Quadrilateral are the solutions of unsolved questions of exercise 8.2 of the NCERT maths text book of class 9.All the solutions of questions are created by future study point by a step by step way for helping the students to boost their preparations for the CBSE board exams.You can also study here NCERT solutions of science and maths from class 9 -12, sample papers, solutions of previous years question papers, tips for entrance exams of government jobs, carrier in online jobs.

Click for online shopping

Future Study Point.Deal: Cloths, Laptops, Computers, Mobiles, Shoes etc

Exercise 8.1- Quadrilateral

You can download pdf of NCERT solutions of class 9 maths chapter 8-Quadrilateral

pdf-NCERT Solutions of Class 9 Maths chapter 8-Quadrilateral

pdf of class 9 NCERT solutions of the chapter 7 -Triangles

NCERT Solutions of class 9 maths

Chapter 1- Number SystemChapter 9-Areas of parallelogram and triangles
Chapter 2-PolynomialChapter 10-Circles
Chapter 3- Coordinate GeometryChapter 11-Construction
Chapter 4- Linear equations in two variablesChapter 12-Heron’s Formula
Chapter 5- Introduction to Euclid’s GeometryChapter 13-Surface Areas and Volumes
Chapter 6-Lines and AnglesChapter 14-Statistics
Chapter 7-TrianglesChapter 15-Probability
Chapter 8- Quadrilateral

CBSE Maths Class IX  solutions of important questions of last years question papers

CBSE Class 9 maths assignment for SA-1

CBSE class 9 Notes on Lines, angles, and triangles

Addition, subtraction, and division of polynomials

Solutions of maths specific questions of mensuration

CBSE class 9 maths, midpoint theorem

NCERT Solutions of Class 9 Science : Chapter 1 to Chapter 15

NCERT Solutions Class 9 Maths Exercise 8.2 of chapter 8- Quadrilateral

Q1.ABCD is a quadrilateral in which P, Q, R, and S midpoints of the sides AB, BC, CD, and DA (see the given figure ).AC is a diagonal. Show that:

Q1. Ex.8.1 class 9 maths

(i) SR ll AC, SR = 1/2(AC)

(ii) PQ = SR

(iii)PQRS is a parallelogram

Ans.

GIVEN: ABCD is a quadrilateral in which P,Q,R and S mid points of the sides AB,BC, CD and DA

TO PROVE:

(i) SR ll AC, SR = 1/2(AC)

(ii) PQ = SR

(iii)PQRS is a parallelogram

PROOF:

(i) In ΔADC, R is the mid point point of DC and S is the mid point of AD

According to mid point theorem the line segment joining the mid points of a triangle is parallel to third side and half of the third side in length.

SR ll AC

SR = 1/2(AC)

(ii) In ΔABC ,P is the mid point of AB and Q is the mid point of  BC ,then using mid point theorem

PQ ll AC

PQ = 1/2(AC)…(i)

SR = 1/2(AC) …(ii)[proved above in (i)]

From equation (i) and (ii)

PQ = SR

(iii) According to mid point theorem,from the figure we have

SR ll AC….(i)

PQ ll AC….(ii)

From equation (i) and (ii) PQ ll SR

PQ = SR (proved above)

According to the theorem of the parallelogram, if one pair of opposite sides of a quadrilateral are equal and parallel, then the other pair of opposite sides is also parallel and equal.

PS ll QR and PS = QR

Therefore PQRS is a parallelogram, Hence proved.

How to choose your subjects after 10 or 12 pass

Q2.ABCD is a rhombus and P, Q, R, and S are midpoints of the sides AB, BC, CD, and DA  respectively. Show that quadrilateral PQRS is a rectangle.

Ans.

Q2.EX 8.2 class 9 maths

GIVEN: ABCD is a rhombus and P,Q,R and S are midpoints of the sides AB, BC, CD, and DA

TO PROVE: PQRS is a rectangle.

PROOF: In ΔABD

PS∥BD (mid point theorem)

∴SU ∥ TO…..(i)

In ΔADC

SR∥AC (mid point theorem)

∴ST ∥ OU….(ii)

From equation (i) and equation (ii),we get that OUST is a parallogram

Therefore ∠ UST = ∠TOU (opposite angle of parallelogram)

∠TOU = 90°(diagonal of rhombus bisect each other at 90°)

∴∠ UST = 90°

Similarly ∠PQR = ∠QRS = ∠QPS =90°

Therefore PQRS is a rectangle, Hence proved

Q3.ABCD is a rectangle and P,Q,R and S are midpoints of the sides AB,BC,CD and DA  respectively .Show that quadrilateral PQRS is a rhombus.

Ans.

Q3.Ex.8.2 class 9 math

 

GIVEN:ABCD is a rectangle P,Q,R and S are midpoints of the sides AB,BC,CD and DA  respectively

TO PROVE: quadrilateral PQRS is a rhombus

PROOF:

In ΔABD, P is the mid point of AB and S is the mid point of AD

Therefore applying mid point theorem

PS∥BD

In ΔBDC, Q is the mid point of BC and R is the mid point of DC

Therefore applying mid point theorem

QR∥BD

In ΔADC, S is the mid point of AD and R is the mid point of DC

Therefore applying mid point theorem

SR∥AC

In ΔABC, P is the mid point of AB and Q is the mid point of BC

Therefore applying mid point theorem

PQ∥AC

AC = BD (diagonal of rectangle)

From equation (i), (ii),(iii) and (iv), we have

PQ = QR = SR = PS

Hence PQRS is a rhombus

Q4. ABCD is a trapizium in which AB∥ DC, BD is a diagonal and  E is the mid point of AD. A line is drawn through E parallel to AB intersecting BC at F(see the given figure). Show that F is the mid point of BC.

Q4 Ex 8.2 class 9 MATHS

 

Ans.

GIVEN: AB∥ DC

E is the mid point of AD

EF∥ AB

TO PROVE:F is the mid point of BC

PROOF: Let EF intersects diagonal BD at G

In ΔABD

EF∥ AB

∴EG ∥ AB

According to converse of mid point theorem ,If E is the mid point of AD and EG ∥ AB, then G will also the mid point of BD

In ΔBDC

GF ∥ DC (DC∥ AB∥EF)

According to converse of mid point theorem ,If G is the mid point of BD and GF ∥ DC, then F will also the mid point of BC

Hence proved

NCERT Solutions Class 9 Maths Exercise 8.2 of chapter 8- Quadrilateral

Q5. In a parallelogram ABCD, E and F are mid point of  the sides AB and CD  respectively (see the figure). Show that the line segments AF and EC trisect the diagonal BD. 

Q5. Ex 8.2 class 9 maths

 

Ans.

GIVEN:ABCD is a parallelogram

E is mid point of  the side AB

F is mid point of  the side CD

TO PROVE: DP = PQ = BQ

PROOF: In ABCD  parallelogram

CF = DC/2….(i)(F is mid point of  the side DC)

AE = AB/2(E is mid point of  the side AB)

AB = DC  (opposite sides of parallelogram )

AE = DC/2….(ii)

From the equations (i) and (ii)

CF = AE

CF ∥ AE

If a pair of opposite sides of a quadrilateral is equal and parallel then another pair of opposite sides is also equal and parallel

So. in AECF quadrilateral

AF ∥ CE

Therefore in ΔDQC,F is the mid point of DC

PF ∥ CQ ( since AF ∥ CE proved above)

According to converse of mid point theorem ,If F is the mid point of DC and PF ∥ CQ, then P will also the mid point of DQ.

∴DP = PQ….(i)

Similarly ,we can consider ΔAPB and can prove

PQ = BQ…..(ii)

From equation (i) and (ii)

DP = PQ = BQ

Hence proved

Please follow us on pintrest

Pintrest future study point

Q6. Show that line segments joining the mid points of the opposite side of a quadrilateral bisect each other.

Ans.

Q6. Ex 8.2 class 9 maths

 

GIVEN: ABCD is a quadrilateral

P ,Q,R and S are the mid points of the sides AB, BC, DC and AD respectively.

TO PROVE: PR and QS bisect each other

PROOF:

In ΔABD, P is the mid point of AB and S is the mid point of AD

Therefore applying mid point theorem

PS∥BD

n ΔBDC, Q is the mid point of BC and R is the mid point of DC

Therefore applying mid point theorem

QR∥BD

From (i) and (ii)

PS = QR

PS ∥ QR (since PS ∥ BD and QR ∥ BD)

If a pair of opposite sides of a quadrilateral is equal and parallel then another pair of opposite sides is also equal and parallel

Therefore SR = PQ , SR ∥ PQ

Now it is clear that PQRS is a parallelogram

Since diagonal of parallelogram bisect each other

Therefore PR and QS bisect each other, Hence proved

Q7. ABC is a triangle right angled at C. A line through the mid point M of hypotenuse AB and parallel to BC intersects AC at D. Show that

(i) D is the mid point of AC

(ii) MD ⊥ AC

Ans.

Q7.Ex.8.2 class 9 maths

GIVEN: ΔABC in which ∠C= 90°

M is mid point of hypotenuse AB

BC∥ MD

TO PROVE:

(i) D is the mid point of AC

According to converse of mid point theorem ,If M is the mid point of AB and BC∥ MD, then D will also the mid point of AC.

(ii) MD ⊥ AC

Since BC∥ MD (given)

∴∠MDC + ∠C = 180° (sum of co-interior angles)

∠MDC + 90°= 180°

∠MDC = 180° – 90° = 90°

Therefore MD ⊥ AC

In ΔAMD and ΔCMD ,we have

AD = CD [D is mid point of AC ,proved above in (i)]

DM = DM (common)

∠ADM = ∠CDM = 90° [proved above in (ii)]

ΔADM ≅ ΔCDM(SAS rule of congruency of triangles)

CM = MA (by CPCT)

Since, M is the mid point of AB

Therefore

Hence proved

NCERT Solutions of Science and Maths for Class 9,10,11 and 12

NCERT Solutions for class 9 maths

Chapter 1- Number SystemChapter 9-Areas of parallelogram and triangles
Chapter 2-PolynomialChapter 10-Circles
Chapter 3- Coordinate GeometryChapter 11-Construction
Chapter 4- Linear equations in two variablesChapter 12-Heron’s Formula
Chapter 5- Introduction to Euclid’s GeometryChapter 13-Surface Areas and Volumes
Chapter 6-Lines and AnglesChapter 14-Statistics
Chapter 7-TrianglesChapter 15-Probability
Chapter 8- Quadrilateral

NCERT Solutions for class 9 science 

Chapter 1-Matter in our surroundingsChapter 9- Force and laws of motion
Chapter 2-Is matter around us pure?Chapter 10- Gravitation
Chapter3- Atoms and MoleculesChapter 11- Work and Energy
Chapter 4-Structure of the AtomChapter 12- Sound
Chapter 5-Fundamental unit of lifeChapter 13-Why do we fall ill ?
Chapter 6- TissuesChapter 14- Natural Resources
Chapter 7- Diversity in living organismChapter 15-Improvement in food resources
Chapter 8- MotionLast years question papers & sample papers

NCERT Solutions for class 10 maths

Chapter 1-Real numberChapter 9-Some application of Trigonometry
Chapter 2-PolynomialChapter 10-Circles
Chapter 3-Linear equationsChapter 11- Construction
Chapter 4- Quadratic equationsChapter 12-Area related to circle
Chapter 5-Arithmetic ProgressionChapter 13-Surface areas and Volume
Chapter 6-TriangleChapter 14-Statistics
Chapter 7- Co-ordinate geometryChapter 15-Probability
Chapter 8-Trigonometry

CBSE Class 10-Question paper of maths 2021 with solutions

CBSE Class 10-Half yearly question paper of maths 2020 with solutions

CBSE Class 10 -Question paper of maths 2020 with solutions

CBSE Class 10-Question paper of maths 2019 with solutions

NCERT Solutions for Class 10 Science

Chapter 1- Chemical reactions and equationsChapter 9- Heredity and Evolution
Chapter 2- Acid, Base and SaltChapter 10- Light reflection and refraction
Chapter 3- Metals and Non-MetalsChapter 11- Human eye and colorful world
Chapter 4- Carbon and its CompoundsChapter 12- Electricity
Chapter 5-Periodic classification of elementsChapter 13-Magnetic effect of electric current
Chapter 6- Life ProcessChapter 14-Sources of Energy
Chapter 7-Control and CoordinationChapter 15-Environment
Chapter 8- How do organisms reproduce?Chapter 16-Management of Natural Resources

NCERT Solutions for class 11 maths

Chapter 1-SetsChapter 9-Sequences and Series
Chapter 2- Relations and functionsChapter 10- Straight Lines
Chapter 3- TrigonometryChapter 11-Conic Sections
Chapter 4-Principle of mathematical inductionChapter 12-Introduction to three Dimensional Geometry
Chapter 5-Complex numbersChapter 13- Limits and Derivatives
Chapter 6- Linear InequalitiesChapter 14-Mathematical Reasoning
Chapter 7- Permutations and CombinationsChapter 15- Statistics
Chapter 8- Binomial Theorem Chapter 16- Probability

CBSE Class 11-Question paper of maths 2015

CBSE Class 11 – Second unit test of maths 2021 with solutions

NCERT solutions for class 12 maths

Chapter 1-Relations and FunctionsChapter 9-Differential Equations
Chapter 2-Inverse Trigonometric FunctionsChapter 10-Vector Algebra
Chapter 3-MatricesChapter 11 – Three Dimensional Geometry
Chapter 4-DeterminantsChapter 12-Linear Programming
Chapter 5- Continuity and DifferentiabilityChapter 13-Probability
Chapter 6- Application of DerivationCBSE Class 12- Question paper of maths 2021 with solutions
Chapter 7- Integrals
Chapter 8-Application of Integrals

Class 12 Solutions of Maths Latest Sample Paper Published by CBSE for 2021-22 Term 2

Class 12 Maths Important Questions-Application of Integrals

Class 12 Maths Important questions on Chapter 7 Integral with Solutions for term 2 CBSE Board 2021-22

Solutions of Class 12 Maths Question Paper of Preboard -2 Exam Term-2 CBSE Board 2021-22

Solutions of class 12  maths question paper 2021 preboard exam CBSE Solution

 

 

 

 

Scroll to Top