Explanation: Let the sides be \(a, b, c\). They must be distinct even integers, and they must satisfy the Triangle Inequality: the sum of the lengths of any two sides must be greater than the length of the third side (\(a+b>c, a+c>b, b+c>a\)). Smallest distinct even integers: 2, 4, 6. Triangle Inequality: \(2+4 = 6\). Fails (must be >). Next distinct even integers: 4, 6, 8. Triangle Inequality: \(4+6 > 8\) (10>8) - OK. \(4+8 > 6\) (12>6) - OK. \(6+8 > 4\) (14>4) - OK. Smallest perimeter = \(4+6+8 = 18\).
