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NCERT Solutions for Exercise 8.2 Class 9 Maths Chapter 8 - Quadrilaterals
NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals Exercise 8.2- Download Free PDF
NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals Exercise 8.2- This chapter takes you into the study of quadrilaterals, four-sided shapes. 9th class maths exercise 8.2 answers provide practical problems and solutions to help you understand their properties and types. Access these free PDF solutions to improve your understanding, prepare for exams, and strengthen your math skills.
The NCERT Solutions for Class 9 Maths Exercise 8.2 deal with concepts that are mostly related to triangles rather than quadrilateral.
The question that is derived using the midpoint theorem from Class 9 Maths Chapter 8 exercise 8.2 are mostly revolving around triangles and their areas.
The two theorems that are present in NCERT solutions for Class 9 Maths chapter 8 exercise 8.2 are:
The line segment joining the midpoints of the two sides of a triangle is parallel to the third side of the triangle (also known as the base of the triangle, generally).
The line segment that is drawn through the midpoint of one side of a triangle, and moves parallel to the third side of the triangle (base) bisects the other side of the triangle into two equal halves.
The figure below shows a triangle ABC and the line EF passes through the midpoints of the sides AB and AC. Since it divides the sides into two AE=EB and AB=2AE
Along with Class 9 Maths, chapter 8 exercise 8.2 the following exercise is also present.
NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals Exercise 8.2
Access Quadrilaterals Class 9 Chapter 8 Exercise: 8.2
Answer:
Given: ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig ). AC is a diagonal.
To prove :
and
Proof: In ACD,
S is the midpoint of DA. (Given)
R is the midpoint of DC. (Given)
By midpoint theorem,
and
Answer:
Given: ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig ). AC is a diagonal.
To prove :
Proof : In ACD,
S is mid point of DA. (Given)
R is mid point of DC. (Given)
By mid point theorem,
and
...................................1
In ABC,
P is mid point of AB. (Given)
Q is mid point of BC. (Given)
By mid point theorem,
and
.................................2
From 1 and 2,we get
and
Thus,
Answer:
Given : ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig ). AC is a diagonal.
To prove : PQRS is a parallelogram.
Proof : In PQRS,
Since,
and
.
So,PQRS is a parallelogram.
Answer:
Given: ABCD is a rhombus in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA . AC, BD are diagonals.
To prove: the quadrilateral PQRS is a rectangle.
Proof: In ACD,
S is midpoint of DA. (Given)
R is midpoint of DC. (Given)
By midpoint theorem,
and
...................................1
In ABC,
P is midpoint of AB. (Given)
Q is mid point of BC. (Given)
By mid point theorem,
and
.................................2
From 1 and 2,we get
and
Thus, and
So,the quadrilateral PQRS is a parallelogram.
Similarly, in BCD,
Q is mid point of BC. (Given)
R is mid point of DC. (Given)
By mid point theorem,
So, QN || LM ...........5
LQ || MN ..........6 (Since, PQ || AC)
From 5 and 6, we get
LMPQ is a parallelogram.
Hence, LMN=
LQN (opposite angles of the parallelogram)
But, LMN= 90 (Diagonals of a rhombus are perpendicular)
so, LQN=90
Thus, a parallelogram whose one angle is right angle,ia a rectangle.Hence,PQRS is a rectangle.
Answer:
Given: ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively.
To prove: the quadrilateral PQRS is a rhombus.
Proof :
In ACD,
S is the midpoint of DA. (Given)
R is the midpoint of DC. (Given)
By midpoint theorem,
and
...................................1
In ABC,
P is the midpoint of AB. (Given)
Q is the midpoint of BC. (Given)
By midpoint theorem,
and
.................................2
From 1 and 2, we get
and
Thus, and
So, the quadrilateral PQRS is a parallelogram.
Similarly, in BCD,
Q is the midpoint of BC. (Given)
R is the midpoint of DC. (Given)
By midpoint theorem,
and
...................5
AC = BD.......................6(diagonals )
From 2, 5 and 6, we get
PQ=QR
Thus, a parallelogram whose adjacent sides are equal is a rhombus. Hence, PQRS is a rhombus.
Answer:
Given: ABCD is a trapezium in which , 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 Fig.
).
To prove: F is the mid-point of BC.
In ABD,
E is the midpoint of AD. (Given)
EG || AB (Given)
By converse of midpoint theorem,
G is the midpoint of BD.
In BCD,
G is mid point of BD. (Proved above)
FG || DC (Given)
By converse of midpoint theorem,
F is the midpoint of BC.
Answer:
Given: In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively
To prove: the line segments AF and EC trisect the diagonal BD.
Proof : In quadrilatweral ABCD,
AB=CD (Given)
(E and F are midpoints of AB and CD)
In quadrilateral AECF,
AE=CF (Given)
AE || CF (Opposite sides of a parallelogram)
Hence, AECF is a parallelogram.
In DCQ,
F is the midpoint of DC. (given )
FP || CQ (AECF is a parallelogram)
By converse of midpoint theorem,
P is the mid point of DQ.
DP= PQ....................1
Similarly,
In ABP,
E is the midpoint of AB. (given )
EQ || AP (AECF is a parallelogram)
By converse of midpoint theorem,
Q is the midpoint of PB.
OQ= QB....................2
From 1 and 2, we have
DP = PQ = QB.
Hence, the line segments AF and EC trisect the diagonal BD.
Answer:
Given: ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA. AC, BD are diagonals.
To prove: the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
Proof: In ACD,
S is the midpoint of DA. (Given)
R is midpoint of DC. (Given)
By midpoint theorem,
and
...................................1
In ABC,
P is the midpoint of AB. (Given)
Q is the midpoint of BC. (Given)
By midpoint theorem,
and
.................................2
From 1 and 2, we get
and
Thus, and
So, the quadrilateral PQRS is a parallelogram and diagonals of a parallelogram bisect each other.
Thus, SQ and PR bisect each other.
Answer:
Given: 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.
To prove :D is mid point of AC.
Proof: In ABC,
M is mid point of AB. (Given)
DM || BC (Given)
By converse of mid point theorem,
D is the mid point of AC.
Answer:
Given: ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallels to BC intersects AC at D.
To prove :
Proof : ADM =
ACB (Corresponding angles)
ADM=
. (
ACB =
)
Hence, .
Answer:
Given: 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.
To prove :
Proof : In ABC,
M is the midpoint of AB. (Given)
DM || BC (Given)
By converse of midpoint theorem,
D is the midpoint of AC i.e. AD = DC.
In AMD and
CMD,
AD = DC (proved above)
ADM =
CDM (Each right angle)
DM = DM (Common)
AMD
CMD (By SAS)
AM = CM (CPCT)
But ,
Hence, .
More About NCERT Solutions for Class 9 Maths Exercise 8.2
NCERT solutions Class 9 Maths exercise 8.2 includes some examples and questions that blends the concepts of midpoints theorem from triangles and then it applies the question to quadrilaterals
Since a quadrilateral is generally a combination of two triangles this becomes a possibility to apply the midpoint theorem in quadrilaterals like a trapezium, or a parallelogram.
The figure shown below shows a trapezium ABCD and then we can see two triangles ABD and BCD thus we have two triangles and the line EF is parallel to the sides AB and CD and passes through the midpoints of AD and BC thus we can apply the midpoint theorem in such scenarios.
What is the basic condition required to make a quadrilateral?
A quadrilateral is formed when we have four non-colinear points in a plane and they are connected by straight lines in such a way that they form a closed four-sided figure.
Also Read| Quadrilaterals Class 9 Notes
Benefits of NCERT Solutions for Class 9 Maths Exercise 8.2
Exercise 8.2 Class 9 Maths, is based on mid-point theorem and its application.
From Class 9 Maths chapter 8 exercise 8.2
Understanding the concepts from Class 9 Maths chapter 8 exercise 8.2 will permit us to comprehend the ideas identified with the various kinds of area and side issue of similarity of figures, ideas which are on the way in Class 10 Maths NCERT syllabus.
Key Features of Exercise 8.2 Class 9 Maths
Practical Problems: 9th class maths exercise 8.2 answers presents practical problems related to quadrilaterals, allowing students to apply their knowledge of their properties and classifications.
Step-by-Step Solutions: The class 9 maths ex 8.2 provides step-by-step solutions for each problem, making it easier for students to follow and understand the problem-solving process.
Enhanced Understanding: It aids in enhancing students' understanding of quadrilaterals by offering a variety of problems and solutions to reinforce concepts.
Exam Preparation: The ex 8.2 class 9 helps students prepare for exams by offering a range of problems that align with the curriculum and potential exam questions.
Practice Material: It serves as valuable practice material for students to improve their problem-solving skills and confidence in working with quadrilaterals.
Free Access: The class 9 ex 8.2 and its solutions are readily accessible for free, allowing students to study and practice at their convenience.
Also, See
NCERT Solutions of Class 10 Subject Wise
Subject Wise NCERT Exemplar Solutions
Frequently Asked Question (FAQs)
NCERT solutions for Class 9 Maths exercise 8.2 is based mid-point theorem and its application.
The sum of all the angles of a quadrilateral that has four sides is 360 and the sum of the angles of a triangle is 180 degrees. A triangle has three sides
The midpoint hypothesis expresses that assuming a line segment is drawn joining the midpoints of any different sides of a triangle it is parallels corresponding to the third side and is equivalent to half of the length of the third side.
It will also pass through the mid-point of the other side.
The area of the smaller triangle formed will be one-fourth of the area of the original triangle.
The new smaller triangle will always be similar to the original triangle if it is formed by the line parallel to the base that passes through the mid-points of the side of the triangle.
A total of four triangles are formed when all the diagonals are drawn inside a quadrilateral.
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Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
Quality Controller
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
Product Manager
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
QA Lead
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
Structural Engineer
A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.
Process Development Engineer
The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.
AWS Solution Architect
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
Azure Administrator
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
Computer Programmer
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
Product Manager
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Information Security Manager
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
Automation Test Engineer
An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.
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