This is a case of similar triangles. The ratio of the corresponding sides of the triangle are in proportion .
[tex]\frac{HJ}{HK}=\frac{HM}{HL}[/tex]
It is true for all combination of the corresponding sides. The one above is the one we will use to answer this problem. Now, let's substitute the known legths of the segments in the given proportion.
[tex]\begin{gathered} \frac{HJ}{HK}=\frac{HM}{HL} \\ HL=HM+ML \\ \frac{2}{HK}=\frac{6}{6+36} \\ \frac{2}{HK}=\frac{6}{42} \end{gathered}[/tex]
By cross multiplication, we can solve for HK
[tex]\begin{gathered} \frac{2}{HK}=\frac{6}{42} \\ 2\cdot42=6\cdot HK \\ 84=6\cdot HK \\ \frac{84}{6}=\frac{6\cdot HK}{6} \\ 14=HK \\ HK=14 \end{gathered}[/tex]
Answer: HK = 14
