The perimeter of an isosceles triangle is $20 \mathrm{cm}$ and if $\mathrm{b}$ is the length of the third side which is distinct from the other two equal sides, find $\mathrm{b}.$

In the following isosceles triangle, in which $\mathrm{AC}=\mathrm{BC}$ and $\mathrm{AB}=\mathrm{BD}=\mathrm{DE}=\mathrm{EF}=\mathrm{FC}=6 \mathrm{cm},$ find $\mathrm{BE}.$

Let $\mathrm{P}$ be the centre of the square constructed on hypotenuse $\mathrm{AC}$ of the right angle triangle $\mathrm{ABC}.$ If $\angle \mathrm{BCA}-\angle \mathrm{BAC}=20°.$ Find the value of $\angle \mathrm{ABP}-\angle \mathrm{CBP}.$

In a trapezium $\mathrm{ABCD},$ where $\mathrm{AB}\Vert \mathrm{CD}$ and diagonals $\mathrm{AC}$ intersects the other diagonal $\mathrm{BD}$ at $\mathrm{O}.$ Which of the following statements are always correct?
(i) $∆\mathrm{AOB}~∆\mathrm{COD}$
(ii) $∆\mathrm{DOA}~∆\mathrm{COB}$
(iii) Area $∆\mathrm{DOA}=$ Area $∆\mathrm{COB}$
(iv) Area $∆\mathrm{AOB}=$ Area $∆\mathrm{COD}$

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.