In triangle ABC, angle C is 90, AB = 8.5, tg A = 15/8. Find AC.

From the graphical representation of the problem, we find that AB is the hypotenuse, AC and BC are legs.
Let’s see what tangent is. The tangent of an acute angle in a right-angled triangle is the ratio of the opposite leg to the adjacent leg, that is, in our problem, tg A = BC / AC. Since, according to the condition of the problem, tg A = 15/8, that is, the sides are not expressed in a numerical value, but tg A = BC / AC, then we express the aspect ratio through the variable x: BC = 15 * x, AC = 8 * x.
Let’s use the Pythagorean theorem: in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
AC ^ 2 + BC ^ 2 = AB ^ 2.
(8 * x) ^ 2 + (15 * x) ^ 2 = 8.5 ^ 2.
64 * x ^ 2 + 225 * x ^ 2 = 72.25.
289 * x ^ 2 = 72.25.
x ^ 2 = 72.25 / 289 = 0.25.
x = √0.25 = 0.5.
Find the AC leg:
8 * 0.5 = 4.
Answer: leg AC = 4.