In a triangle ABC AC = BC, angle C = 120 °, AC = 25√3. Find AB.

In the ABC triangle it is known:

AC = BC;
Angle C = 120 °;
AC = 25√3.
Let’s find the side AB of the triangle ABC.

Decision:

Since the angle С is equal to 120 °, then the angle АСН = angle ВСН = 120 ° / 2 = 60 °.

The height CH divides the angle C in half and the AB side in half.

2) Consider a triangle ACN with a right angle H. The height in an isosceles triangle is perpendicular to AB.

sin 60 ° = AH / AC;

AH = AC * sin a;

Since AB = 2 * AN, then we find the value of AB by the formula:

AB = 2 * AC * sin 60;

Substitute the known values and calculate the AB side.

AB = 2 * 25√3 * √3 / 2 = 25 * √3 * √3 = 25 * √9 = 25 * 3 = 75.

Answer: AB = 75.