prove that the area of an equilateral triangle is equal to square root of 3 divided by 4*a squared where a is the side of the traingle
 

Asked by Aniket | 5th Dec, 2015, 12:45: PM

Expert Answer:

L e t space a space b e space t h e space s i d e 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. 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 equals 1 half cross times b a s e cross times h e i g h t T h e space p e r p e n d i c u l a r space l i n e space f r o m space t h e space v e r t e x 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 i s space a space b i s e c t o r space o f space t h e space b a s e. O b s e r v e space t h e space f o l l o w i n g space f i g u r e space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space
L e t space triangle A B C space b e 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 n d space l e t space apostrophe a apostrophe space b e space t h e space s i d e space o f space t h e space t r i a n g l e T h u s comma space t h e space l i n e space A D space i s space t h e space p e r p e n d i c u l a r space b i s e c t o r space o f space t h e space s i d e space B C. rightwards double arrow triangle A D B space i s space a space r i g h t space t r i a n g l e space a n d space A D equals B D equals a over 2 B y space a p p l y i n g space P y t h a g o r a s comma space w e space h a v e comma A D squared plus B D squared equals A B squared rightwards double arrow A D squared plus open parentheses a over 2 close parentheses squared equals a squared rightwards double arrow A D squared equals a squared minus open parentheses a over 2 close parentheses squared rightwards double arrow A D squared equals open parentheses fraction numerator 4 a squared minus a squared over denominator 4 end fraction close parentheses rightwards double arrow A D squared equals fraction numerator 3 a squared over denominator 4 end fraction rightwards double arrow A D equals fraction numerator square root of 3 a over denominator 2 end fraction A R e a space o f space triangle A B C equals 1 half cross times a cross times fraction numerator square root of 3 a over denominator 2 end fraction equals fraction numerator square root of 3 a squared over denominator 4 end fraction

Answered by Vimala Ramamurthy | 5th Dec, 2015, 02:46: PM