To determine the value of \( x \) that makes 13, 84, and \( x \) a Pythagorean triple, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
Let's assume that \( x \) is the hypotenuse. Then we can write the equation:
\[ x^2 = 13^2 + 84^2 \]
Calculating the squares of 13 and 84:
\[ 13^2 = 169 \] \[ 84^2 = 7056 \]
Now, adding these values:
\[ x^2 = 169 + 7056 = 7225 \]
To find \( x \), take the square root of \( 7225 \):
\[ x = \sqrt{7225} = 85 \]
Thus, if \( 13, 84, x \) is a Pythagorean triple, the value of \( x \) is 85.
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