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Given are the lengths of three sides of a triangle in centimeters. Which of the following triplets can form a right-angled triangle ?

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To determine which of the given triplets can form a right-angled triangle, we will use the Pythagorean theorem. According to this theorem, in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. ### Step-by-step Solution: 1. **Identify the Triplets**: We have three sets of lengths given. Let's denote them as: - Triplet 1: 15 cm, 14 cm, 7 cm - Triplet 2: 40 cm, 41 cm, 12 cm - Triplet 3: 8 cm, 15 cm, 17 cm 2. **Apply the Pythagorean Theorem**: For each triplet, we will check if the sum of the squares of the two smaller sides equals the square of the longest side. 3. **Check Triplet 1 (15, 14, 7)**: - Identify the longest side: 15 cm - Calculate: - \( 14^2 + 7^2 = 196 + 49 = 245 \) - \( 15^2 = 225 \) - Compare: \( 245 \neq 225 \) - Conclusion: This triplet does not form a right-angled triangle. 4. **Check Triplet 2 (40, 41, 12)**: - Identify the longest side: 41 cm - Calculate: - \( 40^2 + 12^2 = 1600 + 144 = 1744 \) - \( 41^2 = 1681 \) - Compare: \( 1744 \neq 1681 \) - Conclusion: This triplet does not form a right-angled triangle. 5. **Check Triplet 3 (8, 15, 17)**: - Identify the longest side: 17 cm - Calculate: - \( 8^2 + 15^2 = 64 + 225 = 289 \) - \( 17^2 = 289 \) - Compare: \( 289 = 289 \) - Conclusion: This triplet forms a right-angled triangle. ### Final Answer: The triplet that can form a right-angled triangle is **8 cm, 15 cm, and 17 cm**.
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