The two sides of the quadrilateral are 1 and 5, and one of the diagonals has length 3

The two sides of the quadrilateral are 1 and 5, and one of the diagonals has length 3 and divides this quadrilateral into two isosceles triangles. What is the smallest perimeter this quadrangle can have?

Let’s denote the vertices of the quadrangle as A, B, C, D;
Let AB = 1. The diagonal from the vertex AC = 3. Since the diagonal cuts off an isosceles triangle, then in triangle ABC the only possible side length is BC = 3, because a triangle with sides 1 and 1 cannot have a third side equal to 3, so how even two sides lying on the same line 1 + 1 = 2 are less than 3;
AD = 5. Since the second triangle cut off by the diagonal ACD is also isosceles, DC can be equal to 3 or 5. Choose 3, since we are looking for a quadrilateral with the smallest perimeter.
Determine the perimeter AB + BC + CD + AD = 1 + 3 + 3 + 5 = 12;
Answer: Smallest perimeter = 12.