To determine whether triangle DEF is a right triangle given its side lengths of 50, 40, and 32, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the longest side (the hypotenuse) is equal to the sum of the squares of the other two sides.
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Identify the longest side. In this case, the longest side is 50.
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We apply the Pythagorean theorem: \[ a^2 + b^2 = c^2 \] where \(c\) is the longest side (50), and \(a\) and \(b\) are the other two sides (40 and 32).
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Calculate the squares of the sides:
- \(50^2 = 2500\)
- \(40^2 = 1600\)
- \(32^2 = 1024\)
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Now, check if the sum of the squares of the two shorter sides equals the square of the longest side: \[ 40^2 + 32^2 = 1600 + 1024 = 2624 \]
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Compare this sum to \(50^2\): \[ 50^2 = 2500 \]
Since \(2624 \neq 2500\), the triangle is not a right triangle.