Find the side of an isosceles triangle if its base is 16 and the angle at the base is 30 degrees.

Let ABC – isosceles triangle AB = BC, AC – base, angle A = 30 degrees.
Let’s draw the height of the ВН. Since ABC is isosceles, the АН is not only the height, but also the median, therefore AH = HC = AC / 2 = 16/2 = 8 conventional units.
Consider a triangle ABН: leg AH = 8 cm, leg ВН lies opposite an angle equal to 30 degrees, so the equality is true:
ВН = AB / 2.
By the Pythagorean theorem, we find the length of the lateral side of the triangle ABC, as well as the hypotenuse of the triangle ABН, AB:
AB ^ 2 = (AB / 2) ^ 2 + AH ^ 2;
AB ^ 2 = AB ^ 2/4 + 8 ^ 2;
AB ^ 2 = (AB ^ 2 + 256) / 4;
4AB ^ 2 = AB ^ 2 + 256;
3AB ^ 2 = 256;
AB ^ 2 = 256/3;
AB = √256 / 3 = 16 / √3 = 16√3 / 3 (conventional units).
Answer: AB = 16√3 / 3 conventional units.