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In an equilateral triangle, the ratio of the square of one altitude to the square of one side is

Asked by nitinkumarsingh.singh.75 | 16th Oct, 2015, 12:47: PM

Expert Answer:

$G i v e n space t h a t space 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. space A D space i s space t h e space a l t i t u d e space f r o m space A. 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 space triangle A B D space a n d space triangle A C D. S i n c e space t h e space a l t i t u d e space f r o m space a 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 t o space t h e space o t h e r space s i d e space b i s e c t s space p e r p e n d i c u l a r a l y comma angle A D B equals angle A D C equals 90 degree angle A B D equals angle A C D equals 60 degree space space space space space open square brackets because 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 comma space a l l space a n g l e s space a r e space 60 degree close square brackets A B equals A C 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 open square brackets because space a l l space t h e 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 close square brackets T h u s comma space b y space A n g l 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 triangle A B D space approximately equal to triangle A C D A l s o comma space T h e space r a t i o space o f A r e a s space o f space c o n g r u e n t space t r i a n g l e s space i s space e q u a l space t o space t h e space r a t i o s space o f space t h e i r space s i d e s. T h u s comma space fraction numerator A r e a space o f space triangle A B D over denominator A r e a space o f space triangle A C D end fraction equals fraction numerator B D squared over denominator C D squared end fraction equals rightwards double arrow B D squared equals C D squared W e space k n o w space t h a t space B C equals B D plus C D rightwards double arrow B C squared equals open parentheses B D plus C D close parentheses squared rightwards double arrow B C squared equals 4 B D squared C o n s i d e r space t h e space r a t i o space o f space s q u a r e space o f space o n e space a l t i t u d e space t o space t h e space s q u a r e space o f space o n e space s i d e equals fraction numerator A D squared over denominator B C squared end fraction F r o m space P y t h a g o r a s space T h e o r e m comma space w e space h a v e comma space A B squared equals A D squared plus B D squared rightwards double arrow A D squared equals A B squared minus B D squared fraction numerator A D squared over denominator B C squared end fraction equals fraction numerator A B squared minus B D squared over denominator B C squared end fraction space space space space space space space space space space equals fraction numerator A B squared over denominator B C squared end fraction minus fraction numerator B D squared over denominator B C squared end fraction space space space space space space space space space space equals 1 minus fraction numerator B D squared over denominator 4 B D squared end fraction space space space space space space space space space space equals 1 minus 1 fourth space space space space space space space space space space equals fraction numerator 4 minus 1 over denominator 4 end fraction space space space space space space space space space space equals 3 over 4$

Answered by Vimala Ramamurthy | 16th Oct, 2015, 03:45: PM

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