To determine which value of \( x \) will make \( (x, 63, 65) \) a Pythagorean triple, we can use the Pythagorean theorem, which states:
\[ a^2 + b^2 = c^2 \]
where \( c \) is the hypotenuse (the largest side). In this case, \( 65 \) will be the hypotenuse, and we have the equation:
\[ x^2 + 63^2 = 65^2 \]
Calculating \( 63^2 \) and \( 65^2 \):
\[ 63^2 = 3969 \] \[ 65^2 = 4225 \]
Now, substituting these values into the equation:
\[ x^2 + 3969 = 4225 \]
Subtracting \( 3969 \) from both sides gives:
\[ x^2 = 4225 - 3969 \] \[ x^2 = 256 \]
Taking the square root of both sides:
\[ x = \sqrt{256} = 16 \]
Thus, the value of \( x \) that will make \( (x, 63, 65) \) a Pythagorean triple is:
Response: x = 16