$\square ABCD$ is a square. If its perimeter is $40$, find $AC + BD$. (in $\sqrt{2}$)

In $\Delta ABC$,$m\angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $\overline{AC}$. If $BD = 2CD$,prove that $AC = 5CD$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $\overline{AC}$. Then,the correspondence $ABC \leftrightarrow \ldots$ between $\Delta ABC$ and $\Delta ADB$ is a similarity.

If $\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$,and $\frac{AB}{4} = \frac{XY}{5}$,then $\frac{BC}{YZ} = \ldots$

The perimeter of rhombus $ABCD$ is $116$. If $AC = 42$, find $BD$.

The perimeter of equilateral $\Delta ABC$ is $24$. Find the length of its altitude. (in $\sqrt{3}$)