Question

Find the missing lengths: HI=7 and LH=9, find LK.

Answer 1

Answer:

LK=12    

Step-by-step explanation:

From the given figure, HI=7 and LH=9, and we know that

$(KH)^{2}=(IH)(HL)$

$(KH)^{2}=7{\times}9$

$(KH)^{2}=63$                             (1)

Now, from ΔKHL, we have

$(OH)^{2}=(OK)(OL)$

and from ΔOKH,

$(KH)^{2}=(OH)^{2}+(OK)^2$

$(OK)^2=(KH)^2-(OH)^2$

$(OK)^2=63-(OK)(OL)$

$(OK)^2+(OK)(OL)=63$

$OK(OK+OL)=63$

$OK(KL)=63$                                 (2)

Also, ΔIKL is similar to ΔOHL, therefore

$\frac{OL}{KL}=\frac{9}{16}$

$\frac{KL-OK}{KL}=\frac{9}{16}$

$\frac{KL-OK}{KL}=\frac{9}{16}$

$1-\frac{OK}{KL}=\frac{9}{16}$

$\frac{7}{16}=\frac{OK}{KL}$

Using equation (2), we get

$\frac{7}{16}=\frac{63}{(KL)^2}$

$(KL)^2=\frac{1008}{7}$

$(KL)^2=144$

$KL=12$

Thus, the value of LK is 12.