In triangle ABC, angle C is 90 degrees, CH-height, angle A 30 degrees, AB = 92. Find AH.

It is necessary to find the length of the segment AH.

Angle A is equal to 30 degrees, by the property of a right-angled triangle, we know that the leg, which lies opposite the angle of 30 degrees, is equal to half of the hypotenuse. It follows that the leg BC is equal to half of the hypotenuse AB, that is, 0.5 * 92 = 46 cm.From the Pythagorean theorem the equality of the sum of the squares of the legs to the square of the hypotenuse, we calculate the leg AC as follows:

AC ^ 2 + BC ^ 2 = AB ^ 2 => AC = √ (AB ^ 2-BC ^ 2) or AC = √ (8464-2116) = 23 √ (12) cm.

Since CH is perpendicular to AB, the triangles ANS and ANS are rectangular.

In the ANS triangle, the AC is the hypotenuse and the CH-leg, lying opposite the angle A equal to 30 degrees, from which it follows that CH is equal to half of the AC or CH = 11.5√ (12) cm.

According to the Pythagorean Theorem in the triangle ANS, we find AH as √ (AC ^ 2-CH ^ 2) that is, AH = √ (6348-1587) = 69 cm