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ABC is a Right Angled Triangle such that AB=AC and bisector of angle C intersects the side AB at D. Prove that AC+AD=BC
Asked by abhinavkumar281 | 09 Sep, 2014, 10:20: PM
answered-by-expert 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 r i g h t space t r i a n g l e space A B C comma space r i g h t space a n g l e d space a t space A.
S i n c e space t h e space t r i a n g l e space i s space a n space i s o s c e l e s space t r i a n g l e comma space w e space h a v e comma space A B equals A C L e t space A B equals A C equals a A l s o space t r i a n g l e space A B C space i s space a space r i g h t space a n g l e d space i s o s c e l e s space t r i a n g l e H e n c e 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 B C squared equals a squared plus a squared rightwards double arrow B C equals a square root of 2 L e t space A D equals b rightwards double arrow B D equals a minus b W e space n e e d space t o space p r o v e space t h a t space A C plus A D equals B C T h a t space i s space w e space n e e d space t o space p r o v e space t h a t comma space a plus b equals a square root of 2 A n g l e space b i s e c t o r space T h e o r e m : space A n space a n g l e space b i s e c t o r space o f space a n space a n g l e space o f space a space t r i a n g l e space d i v i d e s space t h e space o p p o s i t e s i d e space i n space t w o space s e g m e n t s space t h a t space a r e space p r o p o r t i o n a l space t o space t h e space o t h e r space t w o space s i d e s space o f space t h e space t r i a n g l e. T h u s comma space b y space a n g l e space b i s e c t o r space t h e o r e m comma space w e space h a v e comma fraction numerator B D over denominator A D end fraction equals fraction numerator B C over denominator A C end fraction rightwards double arrow fraction numerator a minus b over denominator b end fraction equals fraction numerator a square root of 2 over denominator a end fraction rightwards double arrow a over b minus 1 equals square root of 2 rightwards double arrow a over b equals 1 plus square root of 2 rightwards double arrow b open parentheses 1 plus square root of 2 close parentheses equals a rightwards double arrow b equals fraction numerator a over denominator 1 plus square root of 2 end fraction cross times fraction numerator square root of 2 minus 1 over denominator square root of 2 minus 1 end fraction rightwards double arrow b equals fraction numerator a square root of 2 minus a over denominator 1 end fraction rightwards double arrow a plus b equals a square root of 2 T h u s comma space A C plus A D equals B C
Answered by Vimala Ramamurthy | 10 Sep, 2014, 11:52: AM

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