To determine whether the given side lengths define a right triangle, we can use the Pythagorean theorem, which states that for a triangle with sides of lengths \(a\), \(b\), and \(c\) (where \(c\) is the longest side), the triangle is a right triangle if:
\[ a^2 + b^2 = c^2 \]
Now let's evaluate each set of side lengths:
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For the triangle with sides 4 in, 6 in, 8 in:
- Longest side \(c = 8\)
- Calculate \(a^2 + b^2 = 4^2 + 6^2 = 16 + 36 = 52\)
- Calculate \(c^2 = 8^2 = 64\)
- Conclusion: \(52 \neq 64\) (Not a right triangle)
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For the triangle with sides 8 ft, 15 ft, 17 ft:
- Longest side \(c = 17\)
- Calculate \(a^2 + b^2 = 8^2 + 15^2 = 64 + 225 = 289\)
- Calculate \(c^2 = 17^2 = 289\)
- Conclusion: \(289 = 289\) (Right triangle)
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For the triangle with sides 18 cm, 24 cm, 30 cm:
- Longest side \(c = 30\)
- Calculate \(a^2 + b^2 = 18^2 + 24^2 = 324 + 576 = 900\)
- Calculate \(c^2 = 30^2 = 900\)
- Conclusion: \(900 = 900\) (Right triangle)
Now we can fill in the responses:
- 4 in, 6 in, 8 in: No
- 8 ft, 15 ft, 17 ft: Yes
- 18 cm, 24 cm, 30 cm: Yes
So, the final responses are:
- 4 in, 6 in, 8 in: No
- 8 ft, 15 ft, 17 ft: Yes
- 18 cm, 24 cm, 30 cm: Yes