In an isosceles triangle $\mathrm{ABC},$ side $\mathrm{BC}$ is distinct from rest of the two sides. $\mathrm{CD}$ and $\mathrm{BE}$ are the medians on $\mathrm{AB}$ and $\mathrm{AC},$ respectively. If $\mathrm{W},\mathrm{X},\mathrm{Y}$ and $\mathrm{Z}$ represents distinct areas in the triangle, find the ratio of $\mathrm{W}: \mathrm{X}: \mathrm{Y}: \mathrm{Z}.$

In an isosceles triangle $\mathrm{ABC},$ such that $\mathrm{AC}=\mathrm{BC}$ and $\mathrm{DE}\Vert \mathrm{AB},$ where $\mathrm{D}$ and $\mathrm{E}$ lie on the side $\mathrm{AC}$ and $\mathrm{BC},$ respectively. The area of $\mathrm{DOE}=40{ \mathrm{cm}}^2,$ and area of $∆\mathrm{AOB}=250{ \mathrm{cm}}^2,$ find the area (in ${\mathrm{cm}}^2$) of $\mathrm{AOD}$ and area of $∆\mathrm{ACB},$ respectively.

In a trapezium $\mathrm{ABCD},\mathrm{AB}\Vert \mathrm{CD},\mathrm{AD}$ and $\mathrm{BC}$ are not necessarily equal. If $\mathrm{\alpha }=4{ \mathrm{cm}}^2$ and $\mathrm{\gamma }=9{ \mathrm{cm}}^2.$ Find the value of $\frac{(\mathrm{\beta }+\mathrm{\delta })}{(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }+\mathrm{\delta })}.$

In the following figure, quadrilaterals $\mathrm{ABDC}$ and $\mathrm{APQC}$ are concyclic ones. If $\mathrm{AB}$ and $\mathrm{CD}$ are two parallel lines, then $\mathrm{PBDQ}$ is necessary -