solve:

Asked by araima2001 | 12th Oct, 2015, 12:54: 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.
C o n s t r u c t space a n space a n g l e space b i s e c t o r space f r o m space t h e space v e r t e x space A T h e space a n g l e space b i s e c t o r space f r o m space t h e space v e r t e x space o f space t h e space i s o s c e l e s space t r i a n g l e space i s space t h e space p e r p e n d i c u l a r space b i s e c t o r space o f space t h e space t h i r d space s i d e. T h u s comma space A D space i s space t h e space a n g l e space b i s e c t o r space o f space t h e space a n g l e space angle B A C space a n d space A D space i s space t h e space p e r p e n d i c u l a r space b i s e c t o r space o f space t h e space s i d e space B C. G i v e n space t h a t space A B equals A C equals a angle B A C equals alpha rightwards double arrow angle B A D equals alpha over 2 rightwards double arrow angle A B D equals 90 degree minus alpha over 2 T h u s comma space cos angle A B D equals cos open parentheses 90 degree minus alpha over 2 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator B D over denominator A B end fraction space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator B D over denominator a end fraction T h u s comma space B D equals a c o s open parentheses 90 degree minus alpha over 2 close parentheses rightwards double arrow B D equals a sin alpha over 2 S i n c e space D space i s space t h e space m i d p o i n t space o f space B C comma space w e space h a v e comma B C equals B D plus D C rightwards double arrow B C equals a s i n alpha over 2 plus a s i n alpha over 2 rightwards double arrow B C equals 2 a s i n alpha over 2 space... left parenthesis 1 right parenthesis space space space  B y space a p p l y i n g space P y t h a g o r a s space T h e o r e m comma space w e space h a v e comma A B squared equals A D squared plus B D squared rightwards double arrow A D squared equals A B squared minus B D squared rightwards double arrow A D squared equals A B squared minus B D squared rightwards double arrow A D squared equals a squared minus a squared s i n squared alpha over 2 rightwards double arrow A D squared equals a squared open parentheses 1 minus sin squared alpha over 2 close parentheses rightwards double arrow A D squared equals a squared cos squared alpha over 2 rightwards double arrow A D equals a c o s alpha over 2... left parenthesis 2 right parenthesis A r e a space o f space t h e space t r i a n g l e space A B C equals 1 half cross times b a s e cross times h e i g h t space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 half cross times B C cross times A D... left parenthesis 3 right parenthesis S u b s t i t u t i n g space t h e space v a l u e s space o f space A D space a n d space B C space 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 space i n space e q u a t i o n space left parenthesis 3 right parenthesis comma space w e space h a v e comma A r e a space o f space t h e space t r i a n g l e space A B C equals 1 half cross times 2 a s i n alpha over 2 cross times a c o s alpha over 2 equals a squared over 2 sin alpha H e n c e space t h e space c o r r e c t space o p t i o n space i s space left parenthesis C right parenthesis.

Answered by Vimala Ramamurthy | 13th Oct, 2015, 10:03: AM

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