Question

Find the missing lengths. LK=15 and KH=9, find KI and HI.

Answer 1

QUESTION 1

From Pythagoras Theorem;

$|LH|^2+|KH|^2=|LK|^2$

$\Rightarrow |LH|^2+9^2=15^2$

$\Rightarrow |LH|^2+81=225$

$\Rightarrow |LH|^2=225-81$

$\Rightarrow |LH|^2=144$

$\Rightarrow |LH|=\sqrt{144}$

$\Rightarrow |LH|=12$

Let $|HI|=x$

Then, $|LI|=12+x$

From ΔKLI,

$|KI|^2+15^2=(12+x)^2$

$|KI|^2+225=x^2+24x+144$

$|KI|^2=x^2+24x+144-225$

$|KI|^2=x^2+24x-81...(1)$

From ΔHKI,

$|KI|^2=x^2+9^2$

$|KI|^2=x^2+81...(2)$

Solving equation (1) and (2) simultaneously, we have;

$x^2+81=x^2+24x-81...(1)$

$81+81=24x$

$24x=162$

$x=\frac{27}{4}=6.75$

$\therefore |HI|=6.75$

QUESTION 2

Using equation (2)

$|KI|^2=6.75^2+81$

$|KI|^2=126.5625$

$|KI|=\sqrt{126.5625}$

$|KI|=11.25$