In the Triangle ABD, Using Pythagoras theorem we can write
[tex] AB^2=BD^2+AD^2\\
\\
AB^2=9^2+y^2\\
\\
AB^2=81+y^2\\
\\ [/tex]
In the Triangle ACD using Pythagoras Theorem
[tex] AC^2=AD^2+CD^2\\
\\
AC^2=y^2+7^2\\
\\
AC^2=y^2+49\\
[/tex]
Now in the larger Triangle ABC, we can write
[tex] AB^2+AC^2=BC^2 [/tex]
Now substitute the values from the above equations we get
[tex] 81+y^2+49+y^2=(9+7)^2\\
\\
130+2y^2=16^2\\
\\
130+2y^2=256\\
\\
65+y^2=128\\
\\
y^2=128-65=63\\
\\
y=\sqrt{63} =3\sqrt{7} \\ [/tex]
