In the given figure,P,Q and R are the mid-points of sides AB,BC and AC of triangle ABC respectively. CS is drawn parallel to BA to meet PR produced in S. Prove that QRSC is a parallelogram.

Asked by bis0584 | 2nd Mar, 2015, 12:56: 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.
L e t space A B C space b e space t h e space t r i a n g l e. G i v e n space t h a t space P comma space Q space a n d space R space b e space t h e space m i d space p o i n t s space o f space t h e space s i d e s space A B comma space B C space a n d space C A space r e s p e c t i v e l y. C S space i s space d r a w n space p a r a l l e l space t o space B A space t o space m e e t space P R space p r o d u c e d space i n space S. W e space n e e d space t o space p r o v e space t h a t space Q R S C space i s space a space p a r a l l e log r a m. T h e space l i n e space j o i n i n g space t h e space m i d minus 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 a n d space e q u a l space t o space t h e space h a l f space o f space t h e space l e n g t h space o f space t h e space t h i r d space s i d e. T h e r e f o r e comma space R Q space i s space p a r a l l e l space t o space A B comma space P Q space i s space p a r a l l e l space t o space A C space a n d space P R space i s space p a r a l l e l space t o space B C. A n d space R Q equals 1 half A B comma space P Q equals 1 half A C comma space P R equals 1 half B C R Q space i s space p a r a l l e l space t o space A B space a n d space C S space i s space p a r a l l e l space t o space B A rightwards double arrow R Q parallel to S C S i n c e space P space a n d space R space a r e space t h e space m i d p o i n t s space o f space A B space a n d space A C comma space t h e space l i n e space j o i n i n g space P space a n d space R i s space p a r a l l e l space t o space B C rightwards double arrow S R parallel to B C rightwards double arrow S R parallel to C Q T h u s comma space R Q parallel to S C space a n d space S R parallel to C Q rightwards double arrow Q R S C space i s space a space p a r a l l e log r a m.

Answered by Vimala Ramamurthy | 3rd Mar, 2015, 10:56: AM

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