in a triangle ABC, BM and CN are perpendiculars from B and C respectively on any line passing through A. if L is the mid-point of BC, prove that ML=NL.

Asked by sephali | 28th Feb, 2015, 07:11: 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.

I n space t h e space g i v e n space f i g u r e comma space A B C space i s space a space t r i a n g l e. space B M space a n d space C N space a r e space t w o space p e r p e n d i c u l a r s space f r o m space B space a n d space C space r e s p e c t i v e l y space t o space t h e space l i n e space t h r o u g h space A. A l s o space g i v e n space t h a t space L space i s space t h e space m i d p o i n t space o f space B C. rightwards double arrow B L equals C L I f space t h r e e space p a r a l l e l space l i n e s space c u t space e q u a l space i n t e r c e p t s space o n space a space t r a n s v e r s a l comma space t h e y space w i l l space a l s o space c u t space e q u a l i n t e r c e p t s space o n space o t h e r space t r a n s v e r s a l. B M perpendicular l comma space C N perpendicular l space a n d space O L perpendicular l. therefore B M parallel to O L parallel to C N B C space i s space t h e space t r a n s v e r s a l space c u t t i n g space e q u a l space i n t e r c e p t s comma space B L equals C L T h u s comma space M N space i s space t h e space t r a n s v e r s a l space s u c h space t h a t space M O equals O N 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 s comma space triangle L M O space a n d space triangle L N O. O M equals O N angle L O M equals angle L O N equals 90 degree O L equals O L space space space space left square bracket c o m m o n right square bracket T h u s comma space b y space S i d e minus A n g l e minus S i d e space c r i t e r i o n space o f space c o n g r u e n c e comma space w e space h a v e comma space triangle L M O asymptotically equal to triangle L N O T h e space c o r r e s p o n d i n g space p a r t s space o f space t h e space c o n g r u e n t space t r i a n g l e s space a r e space c o n g r e u n t. T h u s comma space M L equals N L

Answered by Vimala Ramamurthy | 28th Feb, 2015, 09:28: AM

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