In a right-angled triangle ABC, the angle c is 90 degrees, AC = 6, the tangent of the angle A = 2√10 / 3. Find AB.

Since in a right-angled triangle the tangent is the ratio of the opposite leg to the hypotenuse, we get that BC / AC = 2 * √10: 3;

Then we get that BC = 2 * √10: 3 * AC = 2 * √10: 3 * 6 = 12 * √10: 3 = 4 * √10;

Let’s use the Pythagorean theorem. Then we get: AB ^ 2 = AC ^ 2 + BC ^ 2;

Then we get AB = √ (AC ^ 2 + BC ^ 2);

AB = √ (6 ^ 2 + (4 * √10) ^ 2);

AB = √ (36 + 16 * 10);

AB = √ (36 + 160);

AB = √196;

AB = 14.

Thus, we found the hypotenuse of a right-angled triangle AB = 14 cm.

Answer: 14 cm.