let the left side of the apartment be (6+x) ft, and the bottom side be (y+10) ft.

we know

$$(6+x)(y+10) = 450$$

and

$$xy = 180$$

from the first equation

$$\begin{align*} \\

&6y + 10x + 60 + xy = 450 \\

&6y+10x + 60 + 180 = 450 \\

& 6y + 10x = 210\end{align*}$$

from the 2nd equation

$$\begin{align*}

&x = 180/y \\

&6y + 10*180/y = 210 \\

&6y^2 - 210y + 1800 = 0 \\

&y^2 - 35y + 300 = 0 \\

&(y-15)(y-20) = 0 \\

&y = 15\mbox{ or }y = 20 \\

&x = 180/y\mbox{ so } \\

&(x,y) = (12,15)\mbox{ or }(9,20)\end{align*}$$

so there are two layouts that satisfy your requirements

1)

kitchen 6x15

living room 12x15 = 180

bedroom 10x18

total area is 18x25 = 450

2)

kitchen 6x20

living room 9x20 = 180

bedroom 10x15

total area is 15x30 = 450