To solve the equation \( 10^2 + x^2 = 50.5^2 \) using the Pythagorean theorem, we will first calculate \( 10^2 \) and \( 50.5^2 \):
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Calculate \( 10^2 \): \[ 10^2 = 100 \]
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Calculate \( 50.5^2 \): \[ 50.5^2 = 2550.25 \]
Now we can substitute these values into the equation:
\[ 100 + x^2 = 2550.25 \]
Next, solve for \( x^2 \):
\[ x^2 = 2550.25 - 100 \] \[ x^2 = 2450.25 \]
Now, take the square root of both sides to solve for \( x \):
\[ x = \sqrt{2450.25} \] \[ x \approx 49.5 \]
Therefore, the value of \( x \) is approximately \( 49.5 \).