Question

the shortest side of triangle ABC is half of the second side and a third of the longest side. How does the perimeter of the triangle compare to the longest side?

Answer 1

x = the shortest side of triangle ABC ;
y = the second side of triangle ABC;
z = the longest site of triangle ABC ;
P = the perimeter of triangle ABC ;

x = ( 1 / 2 )y;
x = ( 1 / 3 )z;
Then ( 1 / 2 )y = ( 1 / 3 )z;
y = 2z / 3 ;
P = x + y + z = ( 1 / 3)z + 2z / 3 + z = z / 3 + 2z / 3 + z = 3z / 3 + z = z + z = 2z ;
But 2z > z ;
then, P > z ;