To determine which set of side lengths forms a right triangle, we can use the Pythagorean theorem, which states that for a right triangle with sides \( a \), \( b \), and hypotenuse \( c \):
\[ c^2 = a^2 + b^2 \]
We need to check each set of lengths to see if this condition holds.
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Set: 2, 3, 13
- Hypotenuse \( c = 13 \)
- \( 13^2 = 169 \)
- \( 2^2 + 3^2 = 4 + 9 = 13 \) (not equal to 169)
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Set: 4, 6, 10
- Hypotenuse \( c = 10 \)
- \( 10^2 = 100 \)
- \( 4^2 + 6^2 = 16 + 36 = 52 \) (not equal to 100)
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Set: 9, 12, 18
- Hypotenuse \( c = 18 \)
- \( 18^2 = 324 \)
- \( 9^2 + 12^2 = 81 + 144 = 225 \) (not equal to 324)
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Set: 15, 36, 39
- Hypotenuse \( c = 39 \)
- \( 39^2 = 1521 \)
- \( 15^2 + 36^2 = 225 + 1296 = 1521 \) (equal to 1521)
The set 15, 36, 39 forms a right triangle, as it satisfies the Pythagorean theorem.