In an isosceles triangle MNK, the lateral side is √3, the angle at the apex of the wound is 120 degrees. Find Perimeter.

The height h, drawn from the top of the isosceles triangle to the base b, is the median bisecting the base and also the bisector bisecting the top.

Since the angle at the vertex is 120 °, the bisector h divides it into two equal angles, each of which is 120/2 = 60 °.

In a right-angled triangle formed by height h, half base b and side a, half of the base is a leg opposite to an angle of 60 °, side a is the hypotenuse.

The ratio of the opposite leg to the hypotenuse is equal to the sine of the angle, which means:

sin 60 ° = (b / 2) / a.

Hence:

b / 2 = a * sin 60 ° = √3 * √3 / 2 = 3/2;

b = 2 * 3/2 = 3 – the base of this triangle.

Find the perimeter:

P = a + a + b = √3 + √3 + 3 = 3 + 2√3 ≈ 6.46.