In triangle ABC, angle C is 90 degrees; angle A is 30 degrees AB is equal to 36√3 Find the height of CH.

Given: △ ABC, ∠C = 90 °, ∠A = 30 °, AB = 36√3 cm.
Find: CH.
Since ∠С = 90 °, then △ ABC is rectangular. AB – hypotenuse, AC and BC – legs, CH – height.
Behind the property of a right-angled triangle (side opposite an angle of 30 degrees):
ВС = 1/2 AB = 36√3 / 2 = 18√3 (cm).
For the height theorem drawn from the vertex of a right angle:
ВН = ВС ^ 2 / AB = (18√3) ^ 2 / 36√3 = 324 * 3: 36√3 = 9 * 3: √3 = 27 / √3 (cm).
Behind the Pythagorean theorem:
BC ^ 2 = BH ^ 2 + CH ^ 2.
Hence:
CH ^ 2 = BC ^ 2 – BH ^ 2 = (18√3) ^ 2 – (27 / √3) ^ 2 = (324 * 3) – (729/3) = 972 – 243 = 729 (cm).
CH = √729 = 27 (cm).
Answer: CH = 27 cm.