In triangle ABC, angle C is 90 degrees, AB = 5, AC = 4. Find ctg B.

From the graphical representation of the problem, we find that AB is the hypotenuse, AC and BC are legs.
Since ctg B = b / a = BC / AC, you need to find the side BC.
Using the Pythagorean formula, which says that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs, we find the side BC. That is:
c ^ 2 = a ^ 2 + b ^ 2 or in our case AB ^ 2 = AC ^ 2 + BC ^ 2.
Hence:
BC ^ 2 = AB ^ 2 – AC ^ 2.
BC = √ (AB ^ 2 – AC ^ 2).
Let’s substitute the values of the sides into the found formula:
BC = √ (5 ^ 2 – 4 ^ 2) = √ (25 – 16) = √9 = 3.
Since all the data is known, we calculate ctg B:
ctg B = BC / AC = 3/4.

Answer: ctg B = 3/4.