Question

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

Answer 1

LK=12

Explanation:

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

`(KH)^(2)=(IH)(HL)`

`(KH)^(2)=7{*}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

`(OL)/(KL)=(9)/(16)`

`(KL-OK)/(KL)=(9)/(16)`

`(KL-OK)/(KL)=(9)/(16)`

`1-(OK)/(KL)=(9)/(16)`

`(7)/(16)=(OK)/(KL)`

Using equation (2), we get

`(7)/(16)=(63)/((KL)^2)`

`(KL)^2=(1008)/(7)`

`(KL)^2=144`

`KL=12`

Thus, the value of LK is 12.