In a right-angled triangle ABC, the angle is C = 90 degrees. AB = 10 cm, angle BAC

In a right-angled triangle ABC, the angle is C = 90 degrees. AB = 10 cm, angle BAC = 60 degrees. Find the BC and the CD height drawn to the hypothenuse.

The acute angle A is known by condition, we find the angle B:
∠ ABC = 90 ° – ∠ BAC = 30 °.
Opposite this angle lies the leg AC, therefore, it is equal to 1 / 2AB = 5 cm.
We find the leg BC by the Pythagorean theorem:
BC = √ (AB² – AC²) = √ (100 – 25) = √75 = 5√3 (cm).
Consider two right-angled triangles ABC and ACD, the triangles have a common acute angle A, which means they are similar.
From the similarity of the triangles, the aspect ratio follows:
AB / AC = BC / CD → CD = AC * BC / AB = 5 * 5√3 / 10 = 5√3 / 2 (cm).
Answer: the BC leg is 5√3 cm, the CD height is 5√3 / 2 cm.