The converse of the Pythagorean theorem states that if a triangle has side lengths \(a\), \(b\), and \(c\) (where \(c\) is the longest side), and if \(a^2 + b^2 = c^2\), then the triangle is a right triangle.
In this case, we have the side lengths 15, 10, and 12. First, we need to identify the longest side, which is 15. Therefore, we can assign:
- \(a = 10\)
- \(b = 12\)
- \(c = 15\)
Now, we will check whether \(a^2 + b^2 = c^2\):
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Calculate \(a^2\): \[ a^2 = 10^2 = 100 \]
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Calculate \(b^2\): \[ b^2 = 12^2 = 144 \]
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Calculate \(c^2\): \[ c^2 = 15^2 = 225 \]
Now, we check if \(a^2 + b^2 = c^2\): \[ a^2 + b^2 = 100 + 144 = 244 \]
Since \(244 \neq 225\), it follows that \(a^2 + b^2 \neq c^2\).
Therefore, according to the converse of the Pythagorean theorem, a triangle with sides 15, 10, and 12 is not a right triangle.