step 1
In the given smaller right triangle on the bottom
we have that
Applying the Pythagorean Theorem
[tex]z^2=y^2+3^2\text{ ----> equation 1}[/tex]
step 2
In the complete right triangle
we have that
Applying the Pythagorean Theorem
[tex]\begin{gathered} (7+3)^2=x^2+z^2 \\ 10^2=x^2+z^2\text{ -----> equation 2} \end{gathered}[/tex]
step 3
In the right triangle of the top
Applying the Pythagorean Theorem
[tex]x^2=7^2+y^2\text{ -----> equation 3}[/tex]
step 4
Substitute equation 3 in equation 2
[tex]\begin{gathered} 10^2=(7^2+y^2)+z^2 \\ 100=49+y^2+z^2\text{ -----> equation 4} \end{gathered}[/tex]
step 5
substitute equation 1 in equation 4
[tex]\begin{gathered} 100=49+y^2+(y^2+9) \\ solve\text{ for y} \\ 2y^2=100-58 \\ 2y^2=42 \\ y^2=21 \\ y=\sqrt{21} \end{gathered}[/tex]
