To determine whether three given side lengths can form a right triangle, we can apply the Pythagorean theorem, which states that in a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), the relationship \(a^2 + b^2 = c^2\) holds.
We'll evaluate each set of side lengths:
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21, 16, & 12:
- Hypotenuse \(c = 21\)
- Check: \(16^2 + 12^2 = 256 + 144 = 400\)
- \(21^2 = 441\)
- Not a right triangle since \(400 \neq 441\).
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9, 12, & 14:
- Hypotenuse \(c = 14\)
- Check: \(9^2 + 12^2 = 81 + 144 = 225\)
- \(14^2 = 196\)
- Not a right triangle since \(225 \neq 196\).
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30, 24, & 18:
- Hypotenuse \(c = 30\)
- Check: \(24^2 + 18^2 = 576 + 324 = 900\)
- \(30^2 = 900\)
- This IS a right triangle since \(900 = 900\).
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24, 7, & 26:
- Hypotenuse \(c = 26\)
- Check: \(24^2 + 7^2 = 576 + 49 = 625\)
- \(26^2 = 676\)
- Not a right triangle since \(625 \neq 676\).
Concluding, the sides that can form a right triangle are 30, 24, & 18.