One of the angles of an isosceles triangle is 120 °. If the base of a triangle is 18 cm, what is its perimeter?

For convenience of explanation and writing, let us designate this triangle by letters ABC with base BC and obtuse angle A;

Let us lower the height AM from corner A to the side BC. And since the triangle is isosceles, this height will be both the median and the bisector;

Now consider the resulting right-angled triangle ABM, in which the angle B = 30 °, since B = (180 – 120) / 2 = 60/2 = 30 °;

Since AM is the median, it means BM = 18/2 = 9cm;

Knowing the leg and the adjacent angle, we find the hypotenuse of the AВM triangle;

AB = BM / cos30 ° = 9 / √3 / 2 = 18 / √3;

Since the triangle ABC is isosceles it means AB = AC = 18 / √3;

Find the perimeter;

P = 36 / √3 + 18 ≈ 20.78 + 18 = 38.78cm.