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Prove that the difference of any two sides of a triangle is less than the third side?
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Prove that the difference of any two sides of a triangle is less than ...
Given: ∆ABC,
To Prove:
(i) AC ~ AB < BC
(ii) BC ~ AC < AB
(iii) BC ~ AB < AC
Construction: From AC, cut off AD = AB. Join BD.
Proof: In ∆ABD,
AB = AD | By construction
∴ ∠ABD = ∠ADB ...(1)
| Angles opposite to equal sides of a triangle are equal
⇒ ∠ABD = ∠2 ...(2)
Now,
Ext. ∠1 > ∠ABD
| An exterior angle of a triangle is greater than either of its interior opposite angles
⇒ ∠1 > ∠2 ...(3)
| Using (2)
Ext. ∠2 > ∠3 ...(4)
An exterior angle of a triangle is greater than either of its interior opposite angles
From (3) and (4),
∠1 > ∠3
∴ BC > DC
| Side opposite to greater angle is longer
⇒ BC > AC - AD
⇒ BC > AC - AB
| ∵ AD = AB (by construction)
⇒ AC - AB < BC
Similarly, we can show that
BC ~ AC < AB
and BC ~ AB < AC
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Prove that the difference of any two sides of a triangle is less than the third side?
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Prove that the difference of any two sides of a triangle is less than the third side? 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 Prove that the difference of any two sides of a triangle is less than the third side? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that the difference of any two sides of a triangle is less than the third side?.
Prove that the difference of any two sides of a triangle is less than the third side? 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 Prove that the difference of any two sides of a triangle is less than the third side? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that the difference of any two sides of a triangle is less than the third side?.
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