QUESTION 1
From Pythagoras Theorem;
[tex]|LH|^2+|KH|^2=|LK|^2[/tex]
[tex]\Rightarrow |LH|^2+9^2=15^2[/tex]
[tex]\Rightarrow |LH|^2+81=225[/tex]
[tex]\Rightarrow |LH|^2=225-81[/tex]
[tex]\Rightarrow |LH|^2=144[/tex]
[tex]\Rightarrow |LH|=\sqrt{144}[/tex]
[tex]\Rightarrow |LH|=12[/tex]
Let [tex]|HI|=x[/tex]
Then, [tex]|LI|=12+x[/tex]
From ΔKLI,
[tex]|KI|^2+15^2=(12+x)^2[/tex]
[tex]|KI|^2+225=x^2+24x+144[/tex]
[tex]|KI|^2=x^2+24x+144-225[/tex]
[tex]|KI|^2=x^2+24x-81...(1)[/tex]
From ΔHKI,
[tex]|KI|^2=x^2+9^2[/tex]
[tex]|KI|^2=x^2+81...(2)[/tex]
Solving equation (1) and (2) simultaneously, we have;
[tex]x^2+81=x^2+24x-81...(1)[/tex]
[tex]81+81=24x[/tex]
[tex]24x=162[/tex]
[tex]x=\frac{27}{4}=6.75[/tex]
[tex]\therefore |HI|=6.75[/tex]
QUESTION 2
Using equation (2)
[tex]|KI|^2=6.75^2+81[/tex]
[tex]|KI|^2=126.5625[/tex]
[tex]|KI|=\sqrt{126.5625}[/tex]
[tex]|KI|=11.25[/tex]
