Let's find the missing values and the Pythagorean triples for each given pair of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
1. For [tex]\( x = 4 \)[/tex] and [tex]\( y = 3 \)[/tex]:
[tex]\[
\left(x^2 - y^2\right)^2 + \left(2xy\right)^2 = \left(x^2 + y^2\right)^2
\][/tex]
- [tex]\( x^2 - y^2 = 4^2 - 3^2 = 16 - 9 = 7 \)[/tex]
- [tex]\( 2xy = 2 \cdot 4 \cdot 3 = 24 \)[/tex]
- [tex]\( x^2 + y^2 = 4^2 + 3^2 = 16 + 9 = 25 \)[/tex]
The sorted Pythagorean triple is:
[tex]\[
(7, 24, 25)
\][/tex]
2. For [tex]\( x = 5 \)[/tex] and an unknown [tex]\( y \)[/tex]:
Given the Pythagorean triple is [tex]\( (9, 40, 41) \)[/tex].
- The triple components [tex]\( 9, 40, 41 \)[/tex].
The correct [tex]\( y \)[/tex]-value for these components is [tex]\( 4 \)[/tex].
3. For unknown [tex]\( x \)[/tex] and [tex]\( y = 3 \)[/tex]:
Given the Pythagorean triple is [tex]\( (27, 36, 45) \)[/tex].
- The triple components [tex]\( 27, 36, 45 \)[/tex].
The correct [tex]\( x \)[/tex]-value for these components is [tex]\( 6 \)[/tex].
4. For [tex]\( x = 7 \)[/tex] and [tex]\( y = 5 \)[/tex]:
[tex]\[
\left(x^2 - y^2\right)^2 + \left(2xy\right)^2 = \left(x^2 + y^2\right)^2
\][/tex]
- [tex]\( x^2 - y^2 = 7^2 - 5^2 = 49 - 25 = 24 \)[/tex]
- [tex]\( 2xy = 2 \cdot 7 \cdot 5 = 70 \)[/tex]
- [tex]\( x^2 + y^2 = 7^2 + 5^2 = 49 + 25 = 74 \)[/tex]
The sorted Pythagorean triple is:
[tex]\[
(24, 70, 74)
\][/tex]
Putting all this together:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x\text{-value} & y\text{-value} & \text{Pythagorean Triple} \\
\hline
4 & 3 & (7,24,25) \\
\hline
5 & 4 & (9,40,41) \\
\hline
6 & 3 & (27,36,45) \\
\hline
7 & 5 & (24,70,74) \\
\hline
\end{array}
\][/tex]
