In a triangle ABC AC = BC, angle C = 120 degrees, AB = √3. Find the AC.

In a triangle ABC, the sides AC = BC.

Since the sides are equal, it means that the triangle is isosceles.

In an isosceles triangle, the angles are equal, that is, angle a = angle c.

Angle C = 120 °, then angle a + angle b = 180 ° – 120 °;

Angle a + angle c = 60 °;

2 * angle a = 60 °;

Angle a = 30 °;

Angle a = angle c = 30 °;

Base AB = √3.

Let’s find the AC.

Let’s draw the height from point c to the base of AB.

CH – height;

AH = AB / 2 = √3 / 2.

The formula is given:

cos a = AH / AC;

From here we will express AC.

AC = AH / cos a;

Substitute the known values and find AC.

AC = (√3 / 2) / cos 30 = (√3 / 2) / (√3 / 2) = √3 / 2 * 2 / √3 = 1/1 * 1/1 = 1;

Answer: AC = 1.