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If (a, b, c) is a Pythagorean triple, which of the following is also a Pythagorean triple?
- a)(a+1, b+1, c+1)
- b)(2a, 2b, 2c)
- c)(a², b², c²)
- d)(a, b, c²)
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If (a, b, c) is a Pythagorean triple, which of the following is also a...
Given (a,b,c) is a Pythagorean triple, a2 + b2 = c2.
For option B: (2a)2 + (2b)2 = 4a2 + 4b2 = 4(a2 + b2) = 4c2 = (2c)2. Hence (2a, 2b, 2c) is a Pythagorean triple.
The other options do not hold in general: option A would require (a+1)2 + (b+1)2 = (c+1)2, which imposes 2(a+b+1)=2c+1 and is not generally true. Option C would need a4 + b4 = c4, which is false in general. Option D replaces c by c2, giving c2 ≠ (c2)2 except in trivial cases; so it is not generally a Pythagorean triple.
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If (a, b, c) is a Pythagorean triple, which of the following is also a...
Understanding Pythagorean Triples
A Pythagorean triple consists of three positive integers (a, b, c) such that a² + b² = c². The most common example is (3, 4, 5).
Option Analysis
Let's examine each option to determine if it yields a Pythagorean triple.
Option A: (a+1, b+1, c+1)
- For this option, we need to check if (a+1)² + (b+1)² = (c+1)².
- Expanding gives us a² + 2a + 1 + b² + 2b + 1 = c² + 2c + 1.
- This does not simplify to a² + b² = c², hence it is not necessarily a Pythagorean triple.
Option B: (2a, 2b, 2c)
- We check if (2a)² + (2b)² = (2c)².
- This simplifies to 4a² + 4b² = 4c².
- Dividing by 4 gives us a² + b² = c², confirming that (2a, 2b, 2c) is indeed a Pythagorean triple.
Option C: (a², b², c²)
- For this option, we need to check if (a²)² + (b²)² = (c²)².
- This expands to a⁴ + b⁴ = c⁴, which is not equivalent to the original Pythagorean condition.
Option D: (a, b, c²)
- Here, we investigate if a² + b² = (c²)².
- This would require a² + b² = c⁴, which again does not hold true.
Conclusion
Thus, the only option that consistently yields a Pythagorean triple is:
Answer: Option B - (2a, 2b, 2c)
A Pythagorean triple consists of three positive integers (a, b, c) such that a² + b² = c². The most common example is (3, 4, 5).
Option Analysis
Let's examine each option to determine if it yields a Pythagorean triple.
Option A: (a+1, b+1, c+1)
- For this option, we need to check if (a+1)² + (b+1)² = (c+1)².
- Expanding gives us a² + 2a + 1 + b² + 2b + 1 = c² + 2c + 1.
- This does not simplify to a² + b² = c², hence it is not necessarily a Pythagorean triple.
Option B: (2a, 2b, 2c)
- We check if (2a)² + (2b)² = (2c)².
- This simplifies to 4a² + 4b² = 4c².
- Dividing by 4 gives us a² + b² = c², confirming that (2a, 2b, 2c) is indeed a Pythagorean triple.
Option C: (a², b², c²)
- For this option, we need to check if (a²)² + (b²)² = (c²)².
- This expands to a⁴ + b⁴ = c⁴, which is not equivalent to the original Pythagorean condition.
Option D: (a, b, c²)
- Here, we investigate if a² + b² = (c²)².
- This would require a² + b² = c⁴, which again does not hold true.
Conclusion
Thus, the only option that consistently yields a Pythagorean triple is:
Answer: Option B - (2a, 2b, 2c)
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