ABC is a triangle and AD is a median in  triangle ABC to the side BC. Prove that AB/AC=BD/DC.

Asked by Dhruv Sharma | 8th Aug, 2014, 01:33: PM

Expert Answer:

A B C space i s space t r i a n g l e. space D space i s space p o i n t space o n space B C space a n d space A D space i s space m e d i a n. W e space k n o w space t h a t space m e d i a n space d i v i d e s space t r i a n g l e space i n t o space t w o space e q u a l space a r e a s. L e t s space p r o v e space t h i s space f i r s t  L e t s space d r a w space a space p e r p e n d i c u l a r space f r o m space A space t o space B C space w h i c h space m e e t s space a t space E. A D space i s space t h e space m e d i a n comma space s o space B D equals D C A r e a space o f space A D B equals 1 half B D cross times A E A r e a space o f space A D C equals 1 half D C cross times A E  B u t space B D equals D C comma space s o space a r e a s space o f space A D B space a n d space A D C space a r e space e q u a l  S o space w i t h space s a m e space a r e a s space a n d space o n e space o f space t h e space s i d e s space i n space b o t h space t r i a n g l e s space e q a u l. space W e space c a n space s a y space A D B thin space i s space s i m i l a r space t o space A D C  S o space f r o m space r u l e s space o f space s i m i l a r i t y space c o n d i t i o n comma space w e space c a n space w r i t e space fraction numerator A B over denominator B D end fraction equals fraction numerator A C over denominator D C end fraction o r space w e space c a n space r e w r i t e space a s space space fraction numerator A B over denominator A C end fraction equals fraction numerator B D over denominator D C end fraction

Answered by Dharma Teja | 11th Aug, 2014, 10:46: AM