To determine whether a triangle with sides of lengths 15, 10, and 12 is a right triangle using the Converse of the Pythagorean Theorem, we first need to identify the longest side, which is 15 in this case.
According to the Pythagorean theorem, a triangle is a right triangle if the square of the length of the longest side (hypotenuse) is equal to the sum of the squares of the lengths of the other two sides (the legs).
Let's calculate:
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Lengths of the sides: \( a = 10 \), \( b = 12 \), and \( c = 15 \) (where \( c \) is the longest side).
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Calculate \( a^2 + b^2 \): \[ a^2 + b^2 = 10^2 + 12^2 = 100 + 144 = 244 \]
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Calculate \( c^2 \): \[ c^2 = 15^2 = 225 \]
Now, we compare \( a^2 + b^2 \) with \( c^2 \): \[ 244 \neq 225 \]
Since \( a^2 + b^2 \) is not equal to \( c^2 \), the triangle with sides 15, 10, and 12 is not a right triangle.
Therefore, the correct statement is: No, because 244 is not equal to 225.