To determine if a set of three side lengths can be the sides of a right triangle, we can use the Pythagorean theorem. In a right triangle with sides \(a\), \(b\), and hypotenuse \(c\), the theorem states:
\[ a^2 + b^2 = c^2 \]
To apply this, we will check each set of values by identifying the longest side as the potential hypotenuse and testing the relationship.
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For 7.5, 10, and 12.5:
- Longest side: 12.5
- Check: \(7.5^2 + 10^2 = 12.5^2\)
- \(56.25 + 100 = 156.25\)
- \(12.5^2 = 156.25\)
- This set satisfies the Pythagorean theorem. It could be a right triangle.
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For 85, 13, and 84:
- Longest side: 85
- Check: \(13^2 + 84^2 = 85^2\)
- \(169 + 7056 = 7225\)
- \(85^2 = 7225\)
- This set satisfies the Pythagorean theorem. It could be a right triangle.
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For 37, 35, and 12:
- Longest side: 37
- Check: \(35^2 + 12^2 = 37^2\)
- \(1225 + 144 = 1369\)
- \(37^2 = 1369\)
- This set satisfies the Pythagorean theorem. It could be a right triangle.
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For 24, 22, and 40:
- Longest side: 40
- Check: \(24^2 + 22^2 = 40^2\)
- \(576 + 484 = 1600\)
- \(40^2 = 1600\)
- This set satisfies the Pythagorean theorem. It could be a right triangle.
Since all four sets seem to satisfy the conditions to form right triangles, I'll reassess the last option to ensure there’s no oversight, particularly looking at the relationships:
For 24, 22, and 40:
If we check again
- \(24^2 + 22^2\):
- \(576 + 484 = 1060\)
- \(40^2 = 1600\)
This does NOT satisfy the condition \(1060 \neq 1600\).
Thus, the side lengths that could NOT form a right triangle are:
24, 22, and 40.
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