let 'a', 'b' and 'c' represent the length of the sides of the triangle

the lengths of the sides of a triangle are 1:3:4

a : b : c = 1 : 3 : 4

we can write:

a = x

b = 3x

c = 4x

for some x, x>0

we notice that a < b < c since x < 3x < 4x

a is the shortest side

the perimeter of the triangle is 72:

a + b + c = 72

x + 3x + 4x = 72

by solving we find:

x = 9

click here to see the step by step solution of the equation:

a = 9 in

the length of the shortest side is 9 inches.