The height CD is drawn in the right-angled triangle ACB (angle C = 90 degrees).

The height CD is drawn in the right-angled triangle ACB (angle C = 90 degrees). Hypotenuse AB is 10cm, angle CDA = 30 degrees. Find BD.

Most likely there is a typo in the condition, the angle CDA is straight (CD is the height) and we are talking about the angle CBA = 30 °. According to the statement about the leg, which lies opposite an angle of 30 °, we find the AC:
AC = 1/2 * AB = 5 (cm).
By the Pythagorean theorem, we find BC.
BC = √ (AB² – AC²) = √ (100 – 25) = √75 = 5√3 (cm).
Find the height of the CD:
CD = AC * BC / AB = 5 * 5√3 / 10 = 5√3 / 2 (cm).
Consider a right-angled triangle BCD, in which leg CD and hypotenuse BC are known. Find the second leg BD:
BD = √ (BC² – CD²) = √ (75 – 75/4) = √225 / 4 = 15/2 = 7.5 (cm).
Answer: BD length is 7.5 cm.