In a right-angled triangle ABC with a right angle C to the hypotenuse AB, the height CH is drawn.

In a right-angled triangle ABC with a right angle C to the hypotenuse AB, the height CH is drawn. Find the tangent of angle ABC if AC = 8, CH = 4√3 and HB = 12.

First way.

Since CH is the height of the ABC triangle, then the BCН triangle is rectangular, in which the lengths of the two legs are known.

Then tgHBC = CH / BH = 4 * √3 / 12 = √3 / 3.

tgABC = tgHBC = √3 / 3.

Second way.

Let us determine the length of the segment AH. Since CH2 = BH * AH, then AH = CH ^ 2 / BH = 48/12 = 4 cm.

Then AB = 12 + 4 = 16 cm.

By the Pythagorean theorem, BC ^ 2 = AB ^ 2 – AC ^ 2 = 256 – 64 = 192.

BC = 8 * √3 cm.

Then tgABC = AC / BC = 8/8 * √3 = 1 / √3 = √3 / 3.

Answer: The tangent of the angle ABC is √3 / 3.