L and M are the midpoints of the diagonals BD and AC respectively of the quadrilateral ABCD. Through D, draw DE equal and parallel to AB. Show that EC is parallel to LM and is double of it

Asked by Paresh | 10th Dec, 2015, 10:19: AM

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.
A B C D space i s space a space q u a d r i l a t e r a l. space M space i s space t h e space m i d p o i n t space o f space t h e space d i a g o n a l space A C space a n d space L space i s space t h e space m i d p o i n t space o f space t h e d i a g o n a l space B D. D E space i s space d r a w n space p a r a l l e l space t o space A B space a n d space D E equals A B T h e r e f o r e comma space A B E D space i s space a space p a r a l l e log r a m. I n space a space p a r a l l e log r a m comma space d i a g o n a l s space b i s e c t space e a c h space o t h e r. T h u s space A E space p a s s e s space t h r o u g h space t h e space m i d p o i n t space o f space B D. H e n c e space A E space p a s s e s space t h r o u g h space L. S o comma space D i a g o n a l s space A E space a n d space B D space b i s e c t space e a c h space o t h e r. T h a t space i s comma space L space i s space t h e space m i d p o i n t space o f space A E. W e space a l s o space h a v e comma space M space i s space t h e space m i d p o i n t space o f space A C. J o i n space E C. N o w space c o n s i d e r space t h e space t r i a n g l e comma space A C E L space i s space t h e space m i d p o i n t space o f space A E space a n d space M space i s space t h e space m i d p o i n t space o f space A C. T h u s 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 L M 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 C a n d space i s space e q u a l space t o space 1 half A C L M equals 1 half A C rightwards double arrow A C equals 2 L M H e n c e space p r o v e d. space

Answered by Vimala Ramamurthy | 10th Dec, 2015, 11:51: AM