How to prove a triangle equilateral when its three altitudes are given and they are equal??????

Asked by Pratima | 27th Sep, 2015, 12:24: PM

Expert Answer:

G i v e n space t h e space a l t i t u d e s space o f space t h e space t r i a n g l e space a r e space e q u a l. W e space n e e d space t o space p r o v e space t h a t space t h e space t r i a n g l e 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. 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 comma
W e space k n o w space t h a t space t h e space a r e a space o f space t h e space t r i a n g l e space A B C equals 1 half cross times B C cross times A D... left parenthesis 1 right parenthesis S i m i l a r l y comma space i f space A C space i s space t h e space b a s e comma space t h e n space A r e a space o f space t h e space t r i a n g l e space A B C equals 1 half cross times A C cross times B E... left parenthesis 2 right parenthesis A n d comma space i f space A B space i s space t h e space b a s e comma space t h e n space A r e a space o f space t h e space t r i a n g l e space A B C equals 1 half cross times A B cross times C F... left parenthesis 3 right parenthesis 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 comma space a n d space left parenthesis 3 right parenthesis comma space w e space h a v e comma 1 half cross times B C cross times A D equals 1 half cross times A C cross times B E equals 1 half cross times A B cross times C F rightwards double arrow B C cross times A D equals A C cross times B E equals A B cross times C F rightwards double arrow B C equals A C equals A B space space space space open square brackets because A D equals B E equals C F close square brackets rightwards double arrow triangle 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.

Answered by Vimala Ramamurthy | 27th Sep, 2015, 01:40: PM

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