In triangle ABC ∠A = 30 °, ∠В = 60 °, AB = 14√3. Find the height drawn from the vertex of the largest angle of the triangle.

Since the two angles of the triangle ABC are equal to 30 and 60, the third angle is: angle ACB = (180 – 60 – 30) = 90.

Then the largest angle ACB = 90, and the desired height is CH.

The BC leg lies opposite the angle 30, then BC = AB / 2 = 14 * √3 / 2 = 7 * √3 cm.

Determine the area of the triangle ABC.

Saavs = AB * BC * Sin60 / 2 = 14 * √3 * 7 * √3 * √3 / 4 = 147 * √3 / 2 cm2.

Also Savs = AB * CH / 2.

СН = 2 * Saс / AB = (2 * 147 * √3 / 2) / 14 * √3 = 21/2 = 10.5 cm.

Answer: The length of the height is 10.5 cm.