Class 9 Exam > Class 9 Questions > If the diagonals of a parallelogram are equal...
Start Learning for Free
If the diagonals of a parallelogram are equal, then show that it is a rectangle.
?
Verified Answer
If the diagonals of a parallelogram are equal, then show that it is a ...
If the diagonals of a parallelogram are equal, then the parallelogram is a rectangle. This can be proven using the properties of a rectangle and the properties of a parallelogram.
A rectangle is a four-sided polygon with opposite sides that are equal in length and opposite angles that are equal in measure. If the diagonals of a parallelogram are equal, then the opposite angles of the parallelogram are equal in measure.
In a parallelogram, the opposite angles are always equal in measure, regardless of whether the diagonals are equal or not. Therefore, if the diagonals of a parallelogram are equal, then the opposite angles of the parallelogram must also be equal in measure.
Since a rectangle is a parallelogram with opposite sides that are equal in length and opposite angles that are equal in measure, it follows that if the diagonals of a parallelogram are equal, then the parallelogram is a rectangle.
Therefore, if the diagonals of a parallelogram are equal, then it is a rectangle.
This question is part of UPSC exam. View all Class 9 courses
Most Upvoted Answer
If the diagonals of a parallelogram are equal, then show that it is a ...
Understanding the Problem
To prove that a parallelogram with equal diagonals is a rectangle, we will use basic properties of parallelograms and triangles.
Properties of Parallelograms
- A parallelogram is a quadrilateral with opposite sides that are equal and parallel.
- The diagonals of a parallelogram bisect each other.
Equal Diagonals
- Let ABCD be a parallelogram where diagonals AC and BD are equal, i.e., AC = BD.
- Since the diagonals bisect each other, let O be the point of intersection. Therefore, AO = OC and BO = OD.
Triangles and Congruence
- In triangles AOB and COD:
- AO = OC (Diagonals bisect each other)
- BO = OD (Diagonals bisect each other)
- AC = BD (Given)
- By the Side-Side-Side (SSS) congruence criterion, triangles AOB and COD are congruent.
Implication of Congruence
- Since triangles AOB and COD are congruent, we have:
- Angle AOB = Angle COD
- Angle OAB = Angle OCD
Conclusion: Right Angles
- In a parallelogram, if the diagonals are equal, then the angles AOB and COD must be right angles.
- Therefore, all angles in the parallelogram ABCD are right angles.
Final Statement
Since a parallelogram with four right angles is defined as a rectangle, we conclude that if the diagonals of a parallelogram are equal, then it is indeed a rectangle.
To prove that a parallelogram with equal diagonals is a rectangle, we will use basic properties of parallelograms and triangles.
Properties of Parallelograms
- A parallelogram is a quadrilateral with opposite sides that are equal and parallel.
- The diagonals of a parallelogram bisect each other.
Equal Diagonals
- Let ABCD be a parallelogram where diagonals AC and BD are equal, i.e., AC = BD.
- Since the diagonals bisect each other, let O be the point of intersection. Therefore, AO = OC and BO = OD.
Triangles and Congruence
- In triangles AOB and COD:
- AO = OC (Diagonals bisect each other)
- BO = OD (Diagonals bisect each other)
- AC = BD (Given)
- By the Side-Side-Side (SSS) congruence criterion, triangles AOB and COD are congruent.
Implication of Congruence
- Since triangles AOB and COD are congruent, we have:
- Angle AOB = Angle COD
- Angle OAB = Angle OCD
Conclusion: Right Angles
- In a parallelogram, if the diagonals are equal, then the angles AOB and COD must be right angles.
- Therefore, all angles in the parallelogram ABCD are right angles.
Final Statement
Since a parallelogram with four right angles is defined as a rectangle, we conclude that if the diagonals of a parallelogram are equal, then it is indeed a rectangle.
Community Answer
If the diagonals of a parallelogram are equal, then show that it is a ...
Hii
Attention Class 9 Students!
To make sure you are not studying endlessly, EduRev has designed Class 9 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 9.
|
|
Explore Courses for Class 9 exam
|
|
Similar Class 9 Doubts
Top Courses for Class 9View all
If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals?
Question Description
If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals?.
If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals?.
Solutions for If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals? in English & in Hindi are available as part of our courses for Class 9.
Download more important topics, notes, lectures and mock test series for Class 9 Exam by signing up for free.
Here you can find the meaning of If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals? defined & explained in the simplest way possible. Besides giving the explanation of
If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals?, a detailed solution for If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals? has been provided alongside types of If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals? theory, EduRev gives you an
ample number of questions to practice If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals? tests, examples and also practice Class 9 tests.
|
|
Explore Courses for Class 9 exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup