To determine the value of \( x \) that will make \( (x, 63, 65) \) a Pythagorean triple, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side. In this case, 65 is the longest side.
The Pythagorean theorem can be expressed as: \[ x^2 + 63^2 = 65^2 \]
Calculating \( 63^2 \) and \( 65^2 \): \[ 63^2 = 3969 \] \[ 65^2 = 4225 \]
Now, substituting these values into the equation: \[ x^2 + 3969 = 4225 \]
Next, we solve for \( x^2 \): \[ x^2 = 4225 - 3969 \] \[ x^2 = 256 \]
Taking the square root of both sides: \[ x = \sqrt{256} \] \[ x = 16 \]
Thus, \( x = 16 \) will make \( (x, 63, 65) \) a Pythagorean triple.
The correct answer is: x=16
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