How do you solve the side of the triangle?

If you are familiar with the 5 - 12 - 13 right triangle, then you know that this triangle is a multiple of it, since 3 * 13 = 39. This makes the other 2 sides 15 and 36.
If you are not familiar with that concept, then the 2 sides are in a ratio of 12 : 5, which means there exists some number x such that 12x is 1 side and 5x is the other. Using the Pythagorean Theorem,

(5x)² + (12x)² = 39²
25x² + 144x² = 39²
169x² = 39²
13 * 13 * x² = 3 * 13 * 3 * 13
x² = 9
x = 3, etc.

Of course, you can always pick up a calculator when you get to the point
169x² = 39²

Since you haven't been specific about the sides, I can't tell you which value, 15 or 36, is the answer to your question.

If you are trying to work this without trig, you just set up an equality. If you turn the slope into a similar triangle, then you get that the hypotenuse is sqrt(12^2+5^2). So therefore you have BC / 39 = 12 / sqrt(12^2+5^2) and then you just solve for BC by multiplying the left side by 39.

If you want to use trig, then the tan(theta) = 12/5, so theta=arctan(12/5). Then 39*sin(theta)=BC.

In Maths The Pythagorean theorem states

a^2 + b^2 = c^2

where c^2 is Hypotenuse
A^2 and B^2 are the length of other two sides.

If C^2 is given Then the
C^2- A^2=B^2
or
C^2-B^2=A^2