In a right-angled triangle ABC with a hypotenuse AC equal to 12 cm, the height BC is drawn.

In a right-angled triangle ABC with a hypotenuse AC equal to 12 cm, the height BC is drawn. Find CD and DA if angle A is 30 degrees

The BC leg is located opposite the angle A = 30 °. According to the statement about such a leg, we get:
BC = 1/2 * AC = 6 (cm).
The leg and hypotenuse are known, we find the second leg AB according to the Pythagorean theorem.
AB = √ (AC² – BC²) = √ (144 – 36) = √108 = 6√3 (cm).
Consider a right-angled triangle ABD.
BD = 1/2 * AB = 1/2 * 6√3 = 3√3 (cm) (leg opposite an angle of 30 °).
DA = √ (AB² – BD²) = √ (108 – 27) = √81 = 9 (cm).
CD = AC – DA = 12 – 9 = 3 (cm).
Answer: CD is 3 cm, DA is 9 cm.