In an equilateral triangle ABC, AD on on BC is constucted such that BD=1/3 BC. Prove That 9AD^2= 7AB^2

Asked by NISHANT Kashyap | 22nd Aug, 2014, 09:48: PM

Expert Answer:

C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
D r a w space A F space p e r p e n d i c u l a r space t o space B C B y space P y t h a g o r a s space T h e o r e m comma A F squared equals A B squared minus B F squared space space space space space space space equals A B squared minus open parentheses B D plus D F close parentheses squared space space space space space space space space equals A B squared minus B D squared minus D F squared minus 2 B D cross times D F space space space space space space space space space space equals A B squared minus fraction numerator A B squared over denominator 9 end fraction minus D F squared minus fraction numerator 2 A B over denominator 3 end fraction cross times fraction numerator A B over denominator 6 end fraction space space space space space open square brackets sin c e comma space B D equals fraction numerator A B over denominator 3 end fraction comma D F equals fraction numerator A B over denominator 6 end fraction close square brackets space space space space space space space space space space space space equals A B squared minus fraction numerator A B squared over denominator 9 end fraction minus D F squared minus fraction numerator A B squared over denominator 9 end fraction..... left parenthesis 1 right parenthesis A l s o space f r o m space t h e space t r i a n g l e comma space A D F comma A F squared equals A D squared minus D F squared..... left parenthesis 2 right parenthesis F r o m space e q u a t i o n s space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space w e space h a v e comma A D squared minus D F squared equals A B squared minus fraction numerator A B squared over denominator 9 end fraction minus D F squared minus fraction numerator A B squared over denominator 9 end fraction rightwards double arrow A D squared equals A B squared minus fraction numerator 2 A B squared over denominator 9 end fraction rightwards double arrow 9 A D squared equals 9 A B squared minus 2 A B squared rightwards double arrow 9 A D squared equals 7 A B squared

Answered by Vimala Ramamurthy | 25th Aug, 2014, 09:33: AM