prove that the medians of an equilateral triangle are equal.
 
plese prove this in full steps.

Asked by harshmandlik | 29th Aug, 2014, 09:30: AM

Expert Answer:

Consider the following figure.
 
C o n s i d e r space t h e space t r i a n g l e s space A B D space a n d space B C E.  S i n c e space A B C space i s space a n space e q u i l a t e r a l space t r i a n g l e comma space a l l space t h e space a n g l e s space a r e space e q u a l space t o space 60 degree T h e r e f o r e comma space angle A B D equals angle B C E equals 60 degree... left parenthesis 1 right parenthesis S i n c e space A D space a n d space B E space a r e space t h e space m e d i a n s comma space w e space h a v e B D equals C D space a n d space A E equals E C A l s o space s i d e s space o f space a n space e q u i l a t e r a l space t r i a n g l e space a r e space e q u a l comma space a n d space t h e r e f o r e comma w e space h a v e comma space A C equals B C rightwards double arrow A E plus E C equals B D plus C D rightwards double arrow C E equals B D.... left parenthesis 2 right parenthesis S i n c e space t h e space m e d i a n s space f r o m space a n space e q u i l a t e r a l space t r i a n g l e space a r e space p e r p e n d i c u l a r space t o space t h e i r b a s e s comma space w e space h a v e comma angle A D B equals angle B E C equals 90 degree... left parenthesis 3 right parenthesis T h u s comma space f r o m space e q u a t i o n s space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis space a n d space left parenthesis 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 triangle A B D approximately equal to triangle B C E 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 r r e s p o n d i n g space t r i a n g l e s space a r e space e q u a l. T h e r e f o r e comma space w e space h a v e A D equals B E S i m i l a r l y comma space w e space c a n space p r o v e comma space B E equals C F space a n d space h e n c e space t h e space m e d i a n s space o f space a n space e q u a i l a t e r a l t r i a n g l e space a r e space e q u a l.
 

Answered by Vimala Ramamurthy | 30th Aug, 2014, 10:47: AM