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In triangle ABC , if AD is the median , then show that : AB2 + AC2 = 2 ( AD2 + BD2 ) ( AB2 means AB square )

Asked by 13th August 2012, 1:05 PM
Answered by Expert
Answer:

                            

Draw AE perpendicular to BC.

Since angle AED = 900,

Therefore, in triangle ADE

 angle ADE < 900 and angle ADB > 900

Thus, triangle ABD is obtuse angled triangle and triangle ADC is acute angled triangle.

Triangle ABD is obtuse angled at D and AE is perpendicular to BD produced,

Therefore,

AB2 = AD2 + BD2 + 2 BD x DE (i)

Triangle ACD is acute angled at D and AE is perpendicular to BD produced,

Therefore,

AC2 = AD2 + DC2 - 2 DC x DE

 AC2 = AD2 + BD2 - 2 BD x DE  (ii)        (since CD = BD)

Adding (i) and (ii)

AB2 + AC2 = 2AD2 + 2BD2

AB2 +AC2 = 2(AD2 + BD2)

 

 

 

                                                           

 


Answered by Expert 13th August 2012, 5:54 PM
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