e squares be $a,$ $b,$ and $c,$ as shown below:

[asy]
draw((0,0)--(12,0)--(0,5)--cycle);
dot((0,0));
dot((12,0));
dot((0,5));
draw((0,0)--(0,5)--(-5,5)--(-5,0)--cycle);
draw((0,0)--(0,-12)--(12,-12)--(12,0));
draw((0,5)--(5,17)--(17,12)--(12,0)--cycle);
label("$P$",(0,5),NW);
label("$Q$",(0,0),SE);
label("$R$",(12,0),E);

label("$a$",(0,0)--(0,5),W);
label("$b$",(0,0)--(-5,0),S);
label("$c$",(0,5)--(12,0),NE);
[/asy]

The area of each square is equal to the square of its side length, so we have the equations:
\[a^2 + b^2 + c^2 = 338,\]
\[a^2 + b^2 = c^2,\]
\[a^2 + c^2 = b^2.\]

Substituting $c^2$ for $a^2 + b^2$ in the first equation, we get:
\[2(a^2 + c^2) = 338,\]
\[2b^2 = 338,\]
\[b^2 = 169.\]

Therefore, $b = 13,$ and the area of the largest square is $b^2 = \boxed{169}$ square centimeters.The answer is: 169
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Angle PQR is a right angle. The three quadrilaterals shown are squares. The sum of the areas of the three squares is 338 square centimeters. What is the number of square centimeters in the area of the largest square?

[asy]
draw((0,0)--(12,0)--(0,5)--cycle);
dot((0,0));
dot((12,0));
dot((0,5));
draw((0,0)--(0,5)--(-5,5)--(-5,0)--cycle);
draw((0,0)--(0,-12)--(12,-12)--(12,0));
draw((0,5)--(5,17)--(17,12)--(12,0)--cycle);
label("$P$",(0,5),NW);
label("$Q$",(0,0),SE);
label("$R$",(12,0),E);

[/asy]

### Answer:
