In triangle ABC AC = BC, angle C is 120, AB = 2√3. Find AC.

Since, according to the condition, AC = BC, then the triangle ABC is isosceles, therefore the angles CAB and CBA at the base of the triangle are equal to each other.

The angles of a triangle add up to 180.

180 = 2 * CAB + ACB.

2 * CAB = 180 – 120 = 60.

CAB = 60/2 = 30.

Let us omit from point C the perpendicular to AB, which is the height, median and bisector, therefore AD = AB / 2 = 2 * √3 / 2 = √3.

Let’s find the height of CD, triangle ABC.

tgA = CD / AD.

СD = tg30 * АD = (1 / √3) * √3 = 1.

In the ACD triangle, the CD leg is located opposite the angle 30, therefore, the AC hypotenuse is equal to the length of these two legs. AC = 2 * CD = 2 * 1 = 2 cm.

Answer: AC = 2 cm.