In triangle ABC, where angle C = 90 degrees and angle B = 60 degrees, the height CD

In triangle ABC, where angle C = 90 degrees and angle B = 60 degrees, the height CD is drawn. Find AD and BD if BC = 6 cm.

In triangle ABC it is known:

Angle C = 90 °;
Angle B = 60 °;
Height СD;
BC = 6 cm.
Find AD and BD.

Solution:

1) If 2 angles in a right-angled triangle ABC are known, then the third angle is equal to the angle A = 30 °.

2) The height of CD is perpendicular to the hypotenuse AB of the triangle ABC.

3) tg a = BC / AC;

AC = BC / tan a = 6 cm / tan 30 ° = 6 cm / (√3 / 3) = 6 cm * 3 / √3 = 18 cm / √3 = 18 * √3 / √9 cm = 6√3 cm;

4) AB = √ (36 * 3 + 36) = √ (36 * 4) = √36 * √4 = 6 * 2 = 12 cm;

5) BC ^ 2 = BD * AB;

BD = BC ^ 2 / AB = 36/12 = 3 cm;

6) AD = AB – BD = 12 cm – 3 cm = 9 cm.

Answer: AD = 9 cm and BD = 3 cm.