In a triangle ABC AC = BC, angle C is 120 degrees, AB is equal to the root of 3. Find AC.

In triangle ABC we find AC if we know:
AC = BC;
Angle C = 120 °;
AB = √3.
Decision:
1) Since the height of the triangle divides the angle C in half, then the ACN angle in the ACN triangle is 120 ° / 2 = 60 °.
Hence, the angle ACN = 60 °.
2) The height divides the base AB in half, so AH is equal to half of AB.
AH = 1/2 * AB = 1/2 * √3 = √3 / 2;
3) Consider a triangle ACN, where the angle H is straight.
sin C = AH / AC;
Hence, AC = AH / sin C;
Substitute the known values and calculate the AC side.
AC = (√3 / 2) / sin 60 = (√3 / 2) / (√3 / 2) = √3 / 2 * 2 / √3 = √3 / √3 * 2/2 = 1.
Answer: AC = 1.