How to determine the sides of this triangle

Question

Given:

ABC is right triangle
CH - height of hypotenuse
ABC is similar to the triangle ACH and CBH
The area of ABC = 30
AB = 13 

Find:

CH, AC BC

So far all I was able to is to find that CH = 60/13

S = AB * CH / 2
30 = 13 * CH / 2
CH = 30 * 2 / 13 => CH = 60/13

But now I have no idea how to fin AC and BC . I thought applying the Pythagoras theorem however, I don't see how I can do this with the current knowledge. Any help is appreciated.

Answer 1

Note that $Ac^2+BC^2=AB^2=13^2=169$ by Phythagorous. Note that the equation $x^2+y^2=169$ has only one integer solution which is $x=12$ and $y=5$ ( or $x=5$ and $y=12$) . Thus $AC=5$ and $BC=12$ (or $AC=12$ and $BC=5$).