To determine if the lengths 14, 50, and 48 satisfy the Pythagorean Theorem, we need to check if the relationship \(a^2 + b^2 = c^2\) holds, where \(c\) is the longest side (hypotenuse) and \(a\) and \(b\) are the other two sides.
Here, the longest side is 50, so we will assign:
- \(a = 14\)
- \(b = 48\)
- \(c = 50\)
Now we will calculate \(a^2 + b^2\) and \(c^2\):
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Calculate \(a^2\) and \(b^2\): \[ a^2 = 14^2 = 196 \] \[ b^2 = 48^2 = 2304 \]
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Now add these two results: \[ a^2 + b^2 = 196 + 2304 = 2500 \]
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Now calculate \(c^2\): \[ c^2 = 50^2 = 2500 \]
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Now compare \(a^2 + b^2\) to \(c^2\): \[ 2500 = 2500 \]
Since \(a^2 + b^2 = c^2\) holds true, the lengths 14, 50, and 48 do satisfy the Pythagorean Theorem.
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