To determine which of the given sets of numbers can represent the sides of 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) equals the sum of the squares of the lengths of the other two sides. The inequality \( a^2 + b^2 = c^2 \) must hold, where \( c \) is the length of the longest side.
Let's analyze each option.
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2.5, 6, and 6.5
- Hypotenuse = 6.5
- Check: \( 2.5^2 + 6^2 = 6.5^2 \)
- \( 6.25 + 36 = 42.25 \) (which equals \( 42.25 \))
- This set is a right triangle.
-
2, 3, and 5
- Hypotenuse = 5
- Check: \( 2^2 + 3^2 = 5^2 \)
- \( 4 + 9 = 25 \)
- This set is not a right triangle.
-
2, 2, and 5
- Hypotenuse = 5
- Check: \( 2^2 + 2^2 = 5^2 \)
- \( 4 + 4 = 25 \)
- This set is not a right triangle.
-
1.5, 2.5, and 4
- Hypotenuse = 4
- Check: \( 1.5^2 + 2.5^2 = 4^2 \)
- \( 2.25 + 6.25 = 16 \)
- This set is not a right triangle.
The only set that satisfies the condition of a right triangle is 2.5, 6, and 6.5.