In the figure, $X$ is a point in the interior of square $ABCD.AXYZ$ is also a square. If $DY=3cm$ and $AZ=2cm,$ then $BY=$

The correct option is C:$7cm$

In the figure, $ABCD$ and $AXYZ$ are squares.

$DY=3cm,$ $AZ=2cm$

$DZ=DY+YZ$

$=DY+AZ=3+2=5cm$

In $\Delta ADZ,\angle Z={90}^{\circ }$

$A{D}^2=A{Z}^2+D{Z}^2=2^2+5^2$$=4+25=29$

In $\angle ABX,\angle X={90}^{\circ }$

$A{B}^2=A{X}^2+B{X}^2$

$\Rightarrow A{D}^2=A{Z}^2+B{X}^2$$(\because AB=AD,AX=AZ,$ sides of square)

$\Rightarrow 29=2^2+B{X}^2$

$\Rightarrow 29=4+B{X}^2$

$\Rightarrow B{X}^2=29-4=25$

$\Rightarrow BX=5cm$

$\therefore BY=BX+XY=5+2=7cm$ [$\because AXYZ$ is a square, $AX=AZ=2cm$]