Question
Use Euclid's formula to generate a primitive Pythagorean triple using these values of m = 3 and n = 2.
Answers
GPT-4o mini
Euclid's formula for generating primitive Pythagorean triples states that if \( m \) and \( n \) are positive integers with \( m > n \), the sides of the triangle can be computed as follows:
\[
a = m^2 - n^2
\]
\[
b = 2mn
\]
\[
c = m^2 + n^2
\]
Given \( m = 3 \) and \( n = 2 \):
1. Calculate \( a \):
\[
a = m^2 - n^2 = 3^2 - 2^2 = 9 - 4 = 5
\]
2. Calculate \( b \):
\[
b = 2mn = 2 \times 3 \times 2 = 12
\]
3. Calculate \( c \):
\[
c = m^2 + n^2 = 3^2 + 2^2 = 9 + 4 = 13
\]
Thus, the primitive Pythagorean triple generated by \( m = 3 \) and \( n = 2 \) is:
\[
(5, 12, 13)
\]
These values satisfy the Pythagorean theorem: \( 5^2 + 12^2 = 25 + 144 = 169 = 13^2 \).
\[
a = m^2 - n^2
\]
\[
b = 2mn
\]
\[
c = m^2 + n^2
\]
Given \( m = 3 \) and \( n = 2 \):
1. Calculate \( a \):
\[
a = m^2 - n^2 = 3^2 - 2^2 = 9 - 4 = 5
\]
2. Calculate \( b \):
\[
b = 2mn = 2 \times 3 \times 2 = 12
\]
3. Calculate \( c \):
\[
c = m^2 + n^2 = 3^2 + 2^2 = 9 + 4 = 13
\]
Thus, the primitive Pythagorean triple generated by \( m = 3 \) and \( n = 2 \) is:
\[
(5, 12, 13)
\]
These values satisfy the Pythagorean theorem: \( 5^2 + 12^2 = 25 + 144 = 169 = 13^2 \).