A Pythagorean triple consists of three positive integers \(a\), \(b\), and \(c\) such that \(a^2 + b^2 = c^2\). In your case, we have the triple \((20, x, 29)\), where \(c = 29\) (the largest number).
We can set up the equation:
\[ 20^2 + x^2 = 29^2 \]
Calculating \(20^2\) and \(29^2\):
\[ 400 + x^2 = 841 \]
Now, we can isolate \(x^2\):
\[ x^2 = 841 - 400 \] \[ x^2 = 441 \]
Taking the square root of both sides:
\[ x = \sqrt{441} \] \[ x = 21 \]
Since \(x < 29\) is satisfied, the value of \(x\) is:
\[ \boxed{21} \]