In the figure,if $DE \parallel BC,$ find the ratio of $\operatorname{ar}(ADE)$ and $\operatorname{ar}(DECB).$

In $\Delta ABC$,$m\angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. Prove that $\frac{1}{BM^2} = \frac{1}{AB^2} + \frac{1}{BC^2}$.

In rhombus $ABCD$,$AC = 15$ and $BD = 36$. Find the perimeter of rhombus $ABCD$.

$\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow ZXY$. If $AB = 12, BC = 8, CA = 10$ and $ZX = 10$,then $XY + YZ = \ldots$

In $\Delta PQR$,the bisector of $\angle P$ intersects $\overline{QR}$ at $S$. If $PQ : PR = 5 : 4$ and $SR = 5.6 \text{ cm}$,then $QR = \ldots \text{ cm}$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $D$ is a point on $\overline{BC}$. Prove that $AD^{2} + BC^{2} = AC^{2} + BD^{2}$.