Answer:
KI=11.25 and HI=6.75
Step-by-step explanation:
Consider the below figure attached with this question.
According to Pythagoras Theorem:
[tex]base^2+perpendicular^2=hypotenuse^2[/tex]
Use Pythagoras in triangle HKL
[tex]LH^2+KH^2=LK^2[/tex]
[tex](LH)^2+9^2=15^2[/tex]
[tex](LH)^2+81=225[/tex]
[tex](LH)^2=144[/tex]
Taking square root on both sides.
[tex]LH=12[/tex]
Let length of HI be x.
LI = 12+x
Use Pythagoras theorem in 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]
Use Pythagoras theorem in ΔHKI,
[tex](KI)^2=x^2+9^2[/tex]
[tex](KI)^2=x^2+81...(2)[/tex]
From (1) and (2) we get
[tex]x^2+81=x^2+24x-81[/tex]
[tex]24x=162[/tex]
[tex]x=\dfrac{162}{24}=6.75[/tex]
Hence, the measure of HI is 6.75 units.
Substitute x=6.75 in equation (2).
[tex](KI)^2=(6.75)^2+81[/tex]
[tex](KI)^2=126.5625[/tex]
Taking square root on both sides.
[tex]KI=\sqrt{126.5625}[/tex]
[tex]KI=11.25[/tex]
Hence, the measure of KI is 11.25 units.
