To determine whether the given side lengths define a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Specifically, if \( a \) and \( b \) are the lengths of the two shorter sides, and \( c \) is the length of the longest side, then:
\[ a^2 + b^2 = c^2 \]
Let's analyze each set of side lengths:
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Triangle Side Lengths: 3 ft, 4 ft, 6 ft
- Sort the sides: 3 ft, 4 ft, 6 ft (c = 6 ft)
- Check: \( 3^2 + 4^2 = 9 + 16 = 25 \) and \( 6^2 = 36 \)
- Since \( 25 \neq 36 \), NO.
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Triangle Side Lengths: 5 in, 12 in, 13 in
- Sort the sides: 5 in, 12 in, 13 in (c = 13 in)
- Check: \( 5^2 + 12^2 = 25 + 144 = 169 \) and \( 13^2 = 169 \)
- Since \( 169 = 169 \), YES.
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Triangle Side Lengths: 25 cm, 31 cm, 40 cm
- Sort the sides: 25 cm, 31 cm, 40 cm (c = 40 cm)
- Check: \( 25^2 + 31^2 = 625 + 961 = 1586 \) and \( 40^2 = 1600 \)
- Since \( 1586 \neq 1600 \), NO.
Now, we can summarize the results:
- 3 ft, 4 ft, 6 ft: NO
- 5 in, 12 in, 13 in: YES
- 25 cm, 31 cm, 40 cm: NO
So the results are:
- 3 ft, 4 ft, 6 ft: NO
- 5 in, 12 in, 13 in: YES
- 25 cm, 31 cm, 40 cm: NO