Take into account that triangles FGH and GKJ are similar. Then, you can write the following equivalence between the lengths of similar sides:
[tex]\frac{GH}{GK}=\frac{FG}{JG}[/tex]
where,
GH = ?
GK = 6
FG = 7 + 4 = 11
JG = 4
Solve for GH, replace the values of the other parameters and simplify:
[tex]\begin{gathered} GH=(\frac{FG}{JG})\cdot GK \\ GH=(\frac{11}{4})\cdot6 \\ GH=\frac{66}{4}=\frac{33}{2} \end{gathered}[/tex]
Now, take into account that:
GH = GK + KH
Solve for KH, replace the values of GH and GK and simplify:
[tex]\begin{gathered} KH=GH-GK \\ KH=\frac{33}{2}-6 \\ KH=\frac{33-12}{2}=\frac{21}{2}=10\frac{1}{2} \end{gathered}[/tex]
Hence, the lngth of KH is 10 1/2
