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ABCD is a quadrilateral with diagonals AC and BD meeting each other at O show that AC + BD greater than half of AB + BC + CD + DA?
Verified Answer
ABCD is a quadrilateral with diagonals AC and BD meeting each other at...
Let ABCD be a quadrilateral with AC and BD be its diagonals.
In ΔABC,
AB + BC > AC (Sum of any two sides of a triangle is greater than the third side)
In ΔACD,
CD + AD > AC
Adding both the equation, we get
(AB + BC + CD + AD) > AC + AC
⇒ (AB + BC + CD + AD) > 2AC
Similarly, it can be proved hat (AB + BC + CD + AD) > 2BD by taking the triangle ABD and BCD.
Now adding these two results, we get
2 (AB + BC + CD + AD) > 2 (AC + BD)
⇒ AB + BC + CD + AD > AC + BD
This question is part of UPSC exam. View all Class 9 courses
Most Upvoted Answer
ABCD is a quadrilateral with diagonals AC and BD meeting each other at...
Proof:
Given:
- Quadrilateral ABCD with diagonals AC and BD meeting at O
To Prove:
- AC + BD > 1/2(AB + BC + CD + DA)
Proof:
1. In triangle AOB,
- AO + OB > AB (Triangle inequality)
2. In triangle BOC,
- BO + OC > BC (Triangle inequality)
3. In triangle COD,
- CO + OD > CD (Triangle inequality)
4. In triangle DOA,
- DO + OA > DA (Triangle inequality)
5. Adding inequalities from steps 1-4,
- (AO + OB) + (BO + OC) + (CO + OD) + (DO + OA) > AB + BC + CD + DA
6. Simplifying,
- 2(AO + BO + CO + DO) > AB + BC + CD + DA
7. Since AC and BD are diagonals of quadrilateral ABCD,
- AC + BD = 2(OO')
- Where O' is the centroid of the quadrilateral
8. By the triangle inequality theorem,
- OO' < 1/2(ao="" +="" bo="" +="" co="" +="" />
9. Substituting in the previous inequality,
- AC + BD = 2(OO') < ao="" +="" bo="" +="" co="" +="" />
10. Combining steps 6 and 9,
- AC + BD < 2(ao="" +="" bo="" +="" co="" +="" do)="" />< ab="" +="" bc="" +="" cd="" +="" />
11. Therefore,
- AC + BD > 1/2(AB + BC + CD + DA)
Given:
- Quadrilateral ABCD with diagonals AC and BD meeting at O
To Prove:
- AC + BD > 1/2(AB + BC + CD + DA)
Proof:
1. In triangle AOB,
- AO + OB > AB (Triangle inequality)
2. In triangle BOC,
- BO + OC > BC (Triangle inequality)
3. In triangle COD,
- CO + OD > CD (Triangle inequality)
4. In triangle DOA,
- DO + OA > DA (Triangle inequality)
5. Adding inequalities from steps 1-4,
- (AO + OB) + (BO + OC) + (CO + OD) + (DO + OA) > AB + BC + CD + DA
6. Simplifying,
- 2(AO + BO + CO + DO) > AB + BC + CD + DA
7. Since AC and BD are diagonals of quadrilateral ABCD,
- AC + BD = 2(OO')
- Where O' is the centroid of the quadrilateral
8. By the triangle inequality theorem,
- OO' < 1/2(ao="" +="" bo="" +="" co="" +="" />
9. Substituting in the previous inequality,
- AC + BD = 2(OO') < ao="" +="" bo="" +="" co="" +="" />
10. Combining steps 6 and 9,
- AC + BD < 2(ao="" +="" bo="" +="" co="" +="" do)="" />< ab="" +="" bc="" +="" cd="" +="" />
11. Therefore,
- AC + BD > 1/2(AB + BC + CD + DA)
Community Answer
ABCD is a quadrilateral with diagonals AC and BD meeting each other at...
It can be square or rhombus
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ABCD is a quadrilateral with diagonals AC and BD meeting each other at O show that AC + BD greater than half of AB + BC + CD + DA?
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ABCD is a quadrilateral with diagonals AC and BD meeting each other at O show that AC + BD greater than half of AB + BC + CD + DA? 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 ABCD is a quadrilateral with diagonals AC and BD meeting each other at O show that AC + BD greater than half of AB + BC + CD + DA? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABCD is a quadrilateral with diagonals AC and BD meeting each other at O show that AC + BD greater than half of AB + BC + CD + DA?.
ABCD is a quadrilateral with diagonals AC and BD meeting each other at O show that AC + BD greater than half of AB + BC + CD + DA? 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 ABCD is a quadrilateral with diagonals AC and BD meeting each other at O show that AC + BD greater than half of AB + BC + CD + DA? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABCD is a quadrilateral with diagonals AC and BD meeting each other at O show that AC + BD greater than half of AB + BC + CD + DA?.
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