The base of an isosceles triangle and the altitude drawn from one of the congruent sides are equal to 18 cm and 15 cm, respectively. find the lenghts of the sides of the triangle

The answer is 16.3 cm

Take a look at the image.

The area of the triangle (A) is:
A = b * hb / 2
A = a * ha / 2

So:
b * hb / 2 = a * ha / 2
b * hb = a * ha

We have:
b = 18 cm
ha = 15 cm
a = ?
hb = ?

1. Find hb.
According to the image, a is hypotenuse of the right triangle with sides hb and b/2. Use the Pythagorean theorem:
a² = (b/2)² + hb² 

b = 18
a² = (18/2)² + hb² 

a² = (9)² + hb² 
a² = 81 + hb² 
hb² = a² - 81

2. Find a.
b * hb = a * ha

Square both sides
b² * hb² = a² * ha²

b = 18 cm
ha = 15 cm
hb² = a² - 81

(18)² * (a² - 81) = a² * (15)²
324 * (a² - 81) = a² * 225
324a² - 26244 = 225a²
324a² - 225a² = 26244
99a² = 26244
a² = 26244/99
a² = 265.1
a = √265.1
a = 16.3