Find the side of an isosceles triangle if the apex angle is 120 ° and the base is 36 cm.

Let’s call the triangle ABC (AC – base, AB = BC).

In an isosceles triangle, the angles at the base are equal. Angle B = 120, which means angle A = angle C = (180 – 120) / 2 = 30 degrees.

Let’s draw the height of the ВН. In an isosceles triangle, BH is also the median, which means that AH = CH = 36/2 = 18 cm.

Consider a triangle BCH: angle c = 30 degrees, CH = 18 cm.

Let us express the cosine of the angle C = CH / BC.

cosC = 18 / ВС

cos 30 = 18 / ВС

ВС = 18 / cos30 = 18 / (√ 3/2) = 36 / √ 3

Multiply the numerator and denominator by square. root of 3 to get rid of the irrationality of expression.

(36 * √ 3) / 3 = 12 * √3.

Answer: the side edge of the triangle is 12 * √ 3.