To determine which value of \( x \) will make \( (36, x, 85) \) a Pythagorean triple, we need to verify if the equation \( 36^2 + x^2 = 85^2 \) holds true.
First, calculate \( 36^2 \) and \( 85^2 \):
\[ 36^2 = 1296, \] \[ 85^2 = 7225. \]
Now substitute these values into the equation:
\[ 1296 + x^2 = 7225. \]
To solve for \( x^2 \), rearrange the equation:
\[ x^2 = 7225 - 1296. \]
Now, calculate the right side:
\[ x^2 = 7225 - 1296 = 5929. \]
Now, take the square root of both sides:
\[ x = \sqrt{5929} = 77. \]
Therefore, the value of \( x \) that makes \( (36, x, 85) \) a Pythagorean triple is \( \boxed{77} \).