if a,b,c are the sides of a scalene triangle

then comment about the sign of the determinant

 

 a      b       c

 b       c       a 

 c      a        b

 

plzzz. tell me whether it is necessary for all the angles to be acute if it is given that the triangle if scalene

Hi

 

without loss of generality assume that a < b < c  and a , b, c are +ve .

 

On expanding the determinant and simplifying  we get

 

D = 3 a*b*c - ( a³ +b³ + c³ )

 

For all numbers a ,b,c  D is - ve .

 

Its not necessary for a scalene triangle to be accute angled. for example  a triangle with angles 120 , 45, 15 . and many more.

 

Cheers!