In the triangle ABC AC = BC, height CH = 6, cos A = √10 / 10. Find AB.

Let us determine what the sine of the angle A will equal, when from the condition of the assignment we know that the cosine of this angle is √10 / 10 (for this we use one of the trigonometric equations):

sin2 A = 1 – (√10 / 10) 2;

sin2 A = 90/100;

sin A1 = 3√10 / 10;

sin A2 = -3√10 / 10 (does not match the condition).

Let’s define what the tangent of angle A will be equal to:

tg A = 3√10 / 10: √10 / 10 = 3.

Let us determine what the side AH will be equal to:

6: 3 = 2.

So triangle ABC is isosceles, then CH is the median and then AB:

2 * 2 = 4.

Answer: 4 cm.