P, Q and R are respectively , the midpoints of sides BC, CA and AB of a triangle ABC. PR and BQ meets X. CR and PQ meet at Y. Prove that XY = 1/4 BC

 

Asked by Paresh | 11th Dec, 2015, 02:36: 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 i d e r space t h e space t r i a n g l e space A B C. H e r e comma space R space i s space t h e space m i d p o i n t space o f space A B space a n d space Q space i s space t h e space m i d p o i n t space o f space A C. M i d p o i n t space T h e o r e m colon T h e space l i n e space s e g m e n t space c o n n e c t i n g space t h e space m i d p o i n t s space o f space t w o space s i d e s space o f space a space t r i a n g l e space i s space p a r a l l e l space t o space t h e space t h i r d space s i d e space a n d space i s space c o n g r u e n t space t o space o n e space h a l f space o f space t h e space t h i r d space s i d e. H e n c e space b y space m i d p o i n t space t h e o r e m comma space w e space h a v e comma space R Q space i s space p a r a l l e l space t o space B C space a n d R Q equals 1 half B C... left parenthesis 1 right parenthesis A l s o comma space P space a n d space Q space a r e space t h e space m i d p o i n t s space o f space t h e space s i d e s space B C space a n d space A C space r e s p e c t i v e l y. E v e r y space t r i a n g l e space h a s space e x a c t l y space t h r e e space m e d i a n s comma space o n e space f r o m space e a c h space v e r t e x comma space a n d space t h e y space a l l space i n t e r s e c t space e a c h space o t h e r space a t space t h e space t r i a n g l e apostrophe s space c e n t r o i d. T h e r e f o r e comma space B Q space a n d space C R space a r e space t h e space m e d i a n s space a n d space i n t e r s e c t space a t space o n e space p o i n t. S i n c e space B Q space i s space t h e space m e d i a n comma space a n y space p o i n t space o n space t h e space l i n e s space B Q i s space t h e space m i d p o i n t space o f space t h e space l i n e space j o i n i n g space t h e space p o i n t s space B C space a n d space A B space a n d space p a r a l l e l space t o space A C. T h u s comma space X space i s space t h e space m i d p o i n t space o f space P R.  S i n c e space C R space i s space t h e space m e d i a n comma space a n y space p o i n t space o n space t h e space l i n e s space C R i s space t h e space m i d p o i n t space o f space t h e space l i n e space j o i n i n g space t h e space p o i n t s space B C space a n d space C A. T h u s comma space Y space i s space t h e space m i d p o i n t space o f space P Q.  C o n s i d e r space t h e space t r i a n g l e space P Q R comma space H e r e comma space X space i s space t h e space m i d p o i n t space o f space P R space a n d space Y space i s space t h e space m i d p o i n t space o f space P Q. T h u s comma space b y space m i d p o i n t space t h e o r e m comma space w e space h a v e comma space X Y space i s space p a r a l l e l space t o space Q R space a n X Y equals 1 half Q R S u b s t i t u t i n g space t h e space v a l u e space o f space Q R space f r o m space e q u a t i o n space left parenthesis 1 right parenthesis space i n space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma X Y equals 1 half cross times 1 half B C equals 1 fourth B C H e n c e space p r o v e d.

Answered by Vimala Ramamurthy | 12th Dec, 2015, 06:26: AM