In triangle ABC, angle C = 90 degrees, CD-height, angle A = 60 degrees, AB = 8cm. Find AC, BC, AD?

Find angle B in this triangle:
∠ B = 90 ° – ∠ A = 90 ° – 60 ° = 30 °.
The AC leg is located opposite an angle of 30 °, which means AC = 1/2 * AB = 4 (cm).
By the Pythagorean theorem, we find the BC leg:
ВС = √ (AB² – AC²) = √ (64 – 16) = √48 = 4√3 (cm).
The CD height can be found using various formulas:
CD = AC * BC / AB = 4 * 4√3 / 8 = 2√3 (cm);
CD = BC * sin B = BC * sin 30 ° = 4√3 * 1/2 = 2√3 (cm);
CD = AB * sin B * cos B = AB * sin 30 ° * cos 30 ° = 8 * 1/2 * √3 / 2 = 2√3 (cm).
Answer: AC = 4 cm, BC = 4√3 cm, CD = 2√3 cm.