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त्रिभुज

Question
CBSEHHIMAH10010127
Wired Faculty App

किसी समबाहु त्रिभुज ABC कि भुजा BC पर एक बिंदु D इस प्रकार स्थित है कि BD space equals space 1 third space BC है। सिद्ध कीजिए कि 9AD2=7AB2 है।

Solution

समकोण ΔABE और ΔACE में,
          AE = AE    [उभयनिष्ट]
          angle AEB space equals space angle AEC space space space space left square bracket 90 degree right square bracket
और      AB = AC
          increment ABE space approximately equal to space increment ACE
rightwards double arrow       BE = CE
rightwards double arrow       BE space equals space CE space equals space 1 half BC
समकोण ΔADE में
           AD squared space equals space AE squared plus DE squared
rightwards double arrow        AD squared space equals space AE squared plus left parenthesis BE minus BD right parenthesis squared
rightwards double arrow        AD squared equals AE squared plus BE squared plus BD squared minus 2 BD. BE
rightwards double arrow        AD squared space equals space left parenthesis AE squared plus BE squared right parenthesis plus BD squared minus 2 BD. BE
                          open square brackets because space space BD equals 1 third BC space और space BE space equals space CE space equals space 1 half BC close square brackets
rightwards double arrow space space space space space space space space space space AD squared space equals space AB squared plus open parentheses 1 third BC close parentheses squared minus 2 cross times 1 third BC cross times 1 half BC rightwards double arrow space space space space space space space space space space AD squared space equals space AB squared plus 1 over 9 BC squared minus BC squared over 3
rightwards double arrow       AD squared space equals space fraction numerator 9 AB squared plus BC squared minus 3 BC squared over denominator 9 end fraction
rightwards double arrow     9 AD squared space equals space 9 AB squared minus 2 BC squared
और        AB = BC = CA
rightwards double arrow     9 AD squared space equals space 9 AB squared space minus space 2 AB squared
rightwards double arrow      9 AD squared equals 7 AB squared

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