Solve the following math problem efficiently and clearly:

- For simple problems (2 steps or fewer):
Provide a concise solution with minimal explanation.

- For complex problems (3 steps or more):
Use this step-by-step format:

## Step 1: [Concise description]
[Brief explanation and calculations]

## Step 2: [Concise description]
[Brief explanation and calculations]

...

Regardless of the approach, always conclude with:

Therefore, the final answer is: $oxed{answer}$. I hope it is correct.

Where [answer] is just the final number or expression that solves the problem.

Problem: Right triangle $ABC$ has one leg of length 6 cm, one leg of length 8 cm and a right angle at $A$. A square has one side on the hypotenuse of triangle $ABC$ and a vertex on each of the two legs of triangle $ABC$. What is the length of one side of the square, in cm? Express your answer as a common fraction.

[asy]
defaultpen(linewidth(0.8));
size(4cm,4cm);

pair A,B,C;

A=(0,0);
B=(2,3);
C=(7,0);

draw(A--B--C--A);

pair a,b,c,d;

a=(2/3)*B+(1/3)*A;
b=(2/3)*B+(1/3)*C;
c=(1.339,0);
d=(3.65,0);

draw(c--a--b--d);

pair x,y,z;

x=(9/10)*B+(1/10)*A;
z=(14/15)*B+(1/15)*C;
y=(2.12,2.5);

draw(x--y--z);

label("$A$",B,N);
label("$B$",A,SW);
label("$C$",C,SE);

[/asy]
user