let 'a', 'b' and 'c' be the sides of the triangle and a<=b<=c

the perimeter of a triangle is 27 m

a + b + c = 27

the second side is twice the first side

b = 2a

the third side is 2 more than the second side

c = 2 + b

by solving the system of equations

a + b + c = 27

b = 2a

c = 2 + b

we find

a = 5 m

b = 10 m

c = 12 m

click here to see the step by step solution of the system of equations

the length of the longest side is 12 m.