In a right-angled triangle, the leg opposite an angle of 30 degrees is b. find the perimeter of the triangle.

Let us denote by a the length of the second leg, and by b – the length of the hypotenuse of this right-angled triangle.

In the initial data for this task, it is reported that the value of the angle opposite the leg b is 30 °.

Since the sum of the angles of any triangle is 180 °, the value of the angle opposite the leg a is 180 – 90 – 30 = 90 – 30 = 60 °.

Applying the sine theorem, we find a and c:

a = b * sin (60 °) / sin (30 °) = b * (√3 / 2) / (1/2) = b√3;

c = b * sin (90 °) / sin (30 °) = b * 1 / (1/2) = 2b.

Knowing the lengths of all sides, we find the perimeter of the triangle:

a + b + c = b√3 + b + 2b = b√3 + 3b = b * (3 + √3).

Answer: b * (3 + √3).