Question
Given that (5,12,13) is a Pythagorean triple, which of the following is also a Pythagorean triple?
Answers
GPT-4o mini
To determine a Pythagorean triple, we know that it must satisfy the condition \( a^2 + b^2 = c^2 \), where \( c \) is the largest number (the hypotenuse) and \( a \) and \( b \) are the other two sides.
The given Pythagorean triple is (5, 12, 13), which satisfies the equation since:
\[
5^2 + 12^2 = 25 + 144 = 169 = 13^2.
\]
To find another Pythagorean triple, we can use some known methods or generate them by multiplying or using integer combinations of existing triples.
Here’s a simple way to generate new Pythagorean triples using (5, 12, 13):
1. **Scaling**: Multiply (5, 12, 13) by a positive integer \( k \). For example, \( k = 2 \) gives:
\[
(5 \cdot 2, 12 \cdot 2, 13 \cdot 2) = (10, 24, 26).
\]
Verifying:
\[
10^2 + 24^2 = 100 + 576 = 676 = 26^2.
\]
So (10, 24, 26) is a Pythagorean triple.
2. **Using the formula**: For any two integers \( m \) and \( n \) (with \( m > n > 0 \)), you can generate a Pythagorean triple using:
\[
(a, b, c) = (m^2 - n^2, 2mn, m^2 + n^2).
\]
For example, using \( m = 2 \) and \( n = 1 \):
\[
a = 2^2 - 1^2 = 4 - 1 = 3,
\]
\[
b = 2 \cdot 2 \cdot 1 = 4,
\]
\[
c = 2^2 + 1^2 = 4 + 1 = 5.
\]
Thus we have (3, 4, 5), another known Pythagorean triple.
If you provide specific options to consider, I can help you confirm if any of them is a Pythagorean triple.
The given Pythagorean triple is (5, 12, 13), which satisfies the equation since:
\[
5^2 + 12^2 = 25 + 144 = 169 = 13^2.
\]
To find another Pythagorean triple, we can use some known methods or generate them by multiplying or using integer combinations of existing triples.
Here’s a simple way to generate new Pythagorean triples using (5, 12, 13):
1. **Scaling**: Multiply (5, 12, 13) by a positive integer \( k \). For example, \( k = 2 \) gives:
\[
(5 \cdot 2, 12 \cdot 2, 13 \cdot 2) = (10, 24, 26).
\]
Verifying:
\[
10^2 + 24^2 = 100 + 576 = 676 = 26^2.
\]
So (10, 24, 26) is a Pythagorean triple.
2. **Using the formula**: For any two integers \( m \) and \( n \) (with \( m > n > 0 \)), you can generate a Pythagorean triple using:
\[
(a, b, c) = (m^2 - n^2, 2mn, m^2 + n^2).
\]
For example, using \( m = 2 \) and \( n = 1 \):
\[
a = 2^2 - 1^2 = 4 - 1 = 3,
\]
\[
b = 2 \cdot 2 \cdot 1 = 4,
\]
\[
c = 2^2 + 1^2 = 4 + 1 = 5.
\]
Thus we have (3, 4, 5), another known Pythagorean triple.
If you provide specific options to consider, I can help you confirm if any of them is a Pythagorean triple.