To determine the Pythagorean triple generated by using [tex]\( x = 4 \)[/tex] and [tex]\( y = 3 \)[/tex], we can use the following formulas:
[tex]\[ a = x^2 - y^2 \][/tex]
[tex]\[ b = 2 \cdot x \cdot y \][/tex]
[tex]\[ c = x^2 + y^2 \][/tex]
Let's substitute [tex]\( x \)[/tex] and [tex]\( y \)[/tex] with 4 and 3 respectively into these formulas.
1. Calculate [tex]\( a \)[/tex]:
[tex]\[ a = x^2 - y^2 \][/tex]
[tex]\[ a = 4^2 - 3^2 \][/tex]
[tex]\[ a = 16 - 9 \][/tex]
[tex]\[ a = 7 \][/tex]
2. Calculate [tex]\( b \)[/tex]:
[tex]\[ b = 2 \cdot x \cdot y \][/tex]
[tex]\[ b = 2 \cdot 4 \cdot 3 \][/tex]
[tex]\[ b = 2 \cdot 12 \][/tex]
[tex]\[ b = 24 \][/tex]
3. Calculate [tex]\( c \)[/tex]:
[tex]\[ c = x^2 + y^2 \][/tex]
[tex]\[ c = 4^2 + 3^2 \][/tex]
[tex]\[ c = 16 + 9 \][/tex]
[tex]\[ c = 25 \][/tex]
The three numbers forming the Pythagorean triple are:
[tex]\[ (7, 24, 25) \][/tex]
Therefore, the correct answer from the given options is:
[tex]\[ 7, 24, and 25 \][/tex]
