Rotation to slope

[Scriptr]

I am making an Asteroids-like game in OpenGL, using LWJGL. I have a box and control handling. Next step: Motion. I have it set so that left arrow rotates left and right arrow rotates right. Now I need Up arrow to go.

I have it set to go straight if it is headed straight up, down, left, or right:

public void go(){
		float r = -(this.r) % 360;
		r = r < 0 ? 360 + r : r;
		
		float dx, dy;
		
		System.out.println(r);
		if(r == 0){ y += 10; return;}
		if(r == 90){ x += 10; return;}
		if(r == 180){ y -= 10; return;}
		if(r == 270){ x -= 10; return;}
}

but now what if it is at an angle? I need to do several things:

1. Determine slope
2. Convert it to delta Y / delta X
3. Find the equivalent fraction whose numerator and denominator are a and b in a pythagorean triple where c = 10 (or close)

Then I can adjust its position by those amounts

How should I approach this problem?
I am aware that slope is equal to tan(Θ), but I have no clue where to begin making it into a fraction.

I could loop through all possible fractions until I get one that is equivalent and a pythagorean triple, but (correct me if I'm wrong) that wouldn't be fast enough for use in a game (currently 20 ticks per second)

[Scriptr]

I continued to do research, and as my search got less and less specific, I found an example for Flash AS3 (see link). Unless this is helpful to people who come across it later, please delete this thread.