In a triangle ABC AC = CB = 4, angle C is 30. Find the height of AH.

It is known that in the triangle ABC the sides AC and CB are equal, which means that the triangle ABC is isosceles, AC and CB are the lateral sides of the triangle ABC.

The height AH divides this isosceles triangle into two equal right-angled triangles: angle AНС = angle AНВ = 90 °.

The angle C is given to be 30 °. It follows from the axiom that the side opposite to an angle of 30 ° is equal to half of the hypotenuse.

Consider a right-angled triangle AНС: AН and НС are legs, which means AC is hypotenuse. The side opposite the 30 ° angle is the height of AH, which means AH = 1/2 * AC. The AC side is equal to 4 by condition, which means:

AH = 1/2 * 4 = 4/2 = 2.

Answer: 2.