Answer:
LK=12
Step-by-step explanation:
From the given figure, HI=7 and LH=9, and we know that
[tex](KH)^{2}=(IH)(HL)[/tex]
[tex](KH)^{2}=7{\times}9[/tex]
[tex](KH)^{2}=63[/tex] (1)
Now, from ΔKHL, we have
[tex](OH)^{2}=(OK)(OL)[/tex]
and from ΔOKH,
[tex](KH)^{2}=(OH)^{2}+(OK)^2[/tex]
[tex](OK)^2=(KH)^2-(OH)^2[/tex]
[tex](OK)^2=63-(OK)(OL)[/tex]
[tex](OK)^2+(OK)(OL)=63[/tex]
[tex]OK(OK+OL)=63[/tex]
[tex]OK(KL)=63[/tex] (2)
Also, ΔIKL is similar to ΔOHL, therefore
[tex]\frac{OL}{KL}=\frac{9}{16}[/tex]
[tex]\frac{KL-OK}{KL}=\frac{9}{16}[/tex]
[tex]\frac{KL-OK}{KL}=\frac{9}{16}[/tex]
[tex]1-\frac{OK}{KL}=\frac{9}{16}[/tex]
[tex]\frac{7}{16}=\frac{OK}{KL}[/tex]
Using equation (2), we get
[tex]\frac{7}{16}=\frac{63}{(KL)^2}[/tex]
[tex](KL)^2=\frac{1008}{7}[/tex]
[tex](KL)^2=144[/tex]
[tex]KL=12[/tex]
Thus, the value of LK is 12.
