In a right-angled triangle ABC, angle C = 90 degrees, AB = 10 cm, angle BAC = 60 degrees

In a right-angled triangle ABC, angle C = 90 degrees, AB = 10 cm, angle BAC = 60 degrees. Find: A) BC B) the height of CD, drawn to the hypotenuse.

In a right-angled triangle, the sum of the acute angles is 90, then the angle ABC = (90 – BAC) = (90 – 60) = 30.

The AC leg lies opposite the angle 30, then its length is equal to half the length of the hypotenuse AB.

AC = AB / 2 = 10/2 = 5 cm.

In a right-angled triangle ABC, according to the Pythagorean theorem, BC ^ 2 = AB ^ 2 – AC ^ 2 = 100 – 25 = 75.

BC = 5 * √3 cm.

In a right-angled triangle ACD, Sin60 = CD / AC.

СD = АС * Sin60 = 5 * √3 / 2 = 2.5 * √3 cm.

Answer: The BC leg is 5 * √3 cm, the CD height is 2.5 * √3 cm.