Sum of two sides of a triangle is greater than the Third Side


 
 
Concept Explanation
 

Sum of Two Sides of a Triangle is Greater Than The Third Side

Theorem: The sum of any two side of a triangle is greater than the third side.

Given: large A;Delta ABC

To Prove:  large AB+AC>BC,AB+BC>AC;and;BC+AC>AB

Construction:  Produce side BA to D such that AD = AC. Join CD.

Proof:  In large Delta ACD, we have

            AC = AD                             [By construction]

large Rightarrow ;;;angle ADC=angle ACD    [Angles opp. to equal sides are equal]

large Rightarrow ;;;angle ACD=angle ADC

large Rightarrow ;;;angle BCA+angle ACD>angle ADC        [large because ;angle BCA+angle ACD>angle ACD]

large Rightarrow ;;;angle BCD>angle ADC

large Rightarrow ;;;angle BCD>angle BDC                     [large because ;angle ADC=angle BDC]

large Rightarrow ;;;BD>BC                         [large because Side opp. to greater angle is larger]

large Rightarrow ;;;BA+AD>BC

large Rightarrow ;;;BA+AC>BC          [large because ;AC=AD (By construction)]

large Rightarrow ;;;AB+AC>BC

Thus,   large Rightarrow ;;;AB+AC>BC

Similarly, large AB+BC>AC;and;BC+AC>AB

Illustration: Prove that any two sides of a triangle are together greater than twice the median drawn to the third side.

Given:  Delta ABC in which AD is a median.

To Prove: AB+AC>2AD

Construction: Produce AD to E such that AD = DE. Join EC.

Proof: In Delta sADB and EDC, we have

                     AD = DE                            [By construction]

                     BD = DC                         [because D is the mid point of BC]

and,      angle ADB=angle EDC          [ver. opp. angle s  are equal ]

So, by SAS criterion of congruence , we have

              Delta ADBcong Delta EDC

Rightarrow ;;;AB=EC                        [Corresponding parts of congruent triangles are equal right]

Thus,  in Delta AEC, we have

        AC+EC>AE            [because Sum of any two sides of a Delta is greater than the third]

Rightarrow     AC+AB>2AD       [because AD = DE therefore  AE = AD +DE = 2AD and EC = AB]

 

Sample Questions
(More Questions for each concept available in Login)
Question : 1

Of the following which option represents the sides of a triangle?

Right Option : A
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Explanation
Question : 2

In bigtriangleup PQR, S is any point on the side QR. Then which of the following must hold for bigtriangleup PRS,?

Right Option : C
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Explanation
Question : 3

In the given figure,  ABCD is a quadrilateral. Then which of the following statement is true?

Right Option : D
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Explanation
 
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