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If ABCD is a quadrilateral is AB + BC + CD + da is lesser than to AC + BD step by step explanation?
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If ABCD is a quadrilateral is AB + BC + CD + da is lesser than to AC +...
Proof of AB + BC + CD + DA < ac="" +="" />
Given: Quadrilateral ABCD
Proof:
Step 1: Draw Diagonals AC and BD
- Draw the diagonals AC and BD in the quadrilateral ABCD.
Step 2: Divide AC and BD into segments
- Divide AC into two segments as AB and BC, and BD into two segments as CD and DA.
Step 3: Apply Triangle Inequality Theorem
- According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
- Applying this theorem to triangles ABC, BCD, CDA, and DAB, we get:
- AB + BC > AC (from triangle ABC)
- BC + CD > BD (from triangle BCD)
- CD + DA > AC (from triangle CDA)
- DA + AB > BD (from triangle DAB)
Step 4: Combine the Inequalities
- Adding all the inequalities obtained in step 3, we get:
- AB + BC + BC + CD + CD + DA + DA + AB > AC + BD
- 2(AB + BC + CD + DA) > AC + BD
- AB + BC + CD + DA > 0.5(AC + BD)
Step 5: Conclusion
- Therefore, the sum of the lengths of the sides of the quadrilateral ABCD (AB + BC + CD + DA) is less than half the sum of the lengths of the diagonals AC and BD (AC + BD).
Given: Quadrilateral ABCD
Proof:
Step 1: Draw Diagonals AC and BD
- Draw the diagonals AC and BD in the quadrilateral ABCD.
Step 2: Divide AC and BD into segments
- Divide AC into two segments as AB and BC, and BD into two segments as CD and DA.
Step 3: Apply Triangle Inequality Theorem
- According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
- Applying this theorem to triangles ABC, BCD, CDA, and DAB, we get:
- AB + BC > AC (from triangle ABC)
- BC + CD > BD (from triangle BCD)
- CD + DA > AC (from triangle CDA)
- DA + AB > BD (from triangle DAB)
Step 4: Combine the Inequalities
- Adding all the inequalities obtained in step 3, we get:
- AB + BC + BC + CD + CD + DA + DA + AB > AC + BD
- 2(AB + BC + CD + DA) > AC + BD
- AB + BC + CD + DA > 0.5(AC + BD)
Step 5: Conclusion
- Therefore, the sum of the lengths of the sides of the quadrilateral ABCD (AB + BC + CD + DA) is less than half the sum of the lengths of the diagonals AC and BD (AC + BD).
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If ABCD is a quadrilateral is AB + BC + CD + da is lesser than to AC + BD step by step explanation?
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If ABCD is a quadrilateral is AB + BC + CD + da is lesser than to AC + BD step by step explanation? for Class 7 2024 is part of Class 7 preparation. The Question and answers have been prepared according to the Class 7 exam syllabus. Information about If ABCD is a quadrilateral is AB + BC + CD + da is lesser than to AC + BD step by step explanation? covers all topics & solutions for Class 7 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If ABCD is a quadrilateral is AB + BC + CD + da is lesser than to AC + BD step by step explanation?.
If ABCD is a quadrilateral is AB + BC + CD + da is lesser than to AC + BD step by step explanation? for Class 7 2024 is part of Class 7 preparation. The Question and answers have been prepared according to the Class 7 exam syllabus. Information about If ABCD is a quadrilateral is AB + BC + CD + da is lesser than to AC + BD step by step explanation? covers all topics & solutions for Class 7 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If ABCD is a quadrilateral is AB + BC + CD + da is lesser than to AC + BD step by step explanation?.
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