In a right-angled triangle ABC, the hypotenuse AC is 12 cm. BD is the height of this triangle.

In a right-angled triangle ABC, the hypotenuse AC is 12 cm. BD is the height of this triangle. Find CD and DA if the angle is A = 30 degrees.

1. The BC leg is opposite an angle of 30º, therefore its length is 1/2 AB:

BC = AB / 2 = 12/2 = 6 cm.

2. AB = √AC ^ 2 – BC ^ 2 (according to the Pythagorean theorem);

AB = √12 ^ 2 – 6 ^ 2 = √144 – 36 = √108 = √36 x 3 = 6√3 cm.

3. Leg BD is opposite an angle of 30º, therefore its length is 1/2 AB:

BD = AB / 2 = 6√3 / 2 = 3√3 cm.

4. AD ^ 2 = AB ^ 2 – BD ^ 2 (according to the Pythagorean theorem);

AD ^ 2 = (6√3) ^ 2 – (3√3) ^ 2 = 36 x 3 – 9 x 3 = 108 – 27 = 81;

AD = √81 = 9 cm.

5.CD = 12 – 9 = 3 cm.

Answer: the length of the CD is 3 cm, the length of the AD is 9 cm.