Two right triangles $\mathrm{ABC}$ and $\mathrm{DBC}$ are drawn on the same hypotenuse $\mathrm{BC}$ on the same side of $\mathrm{BC}.$ If $\mathrm{AC}$ and $\mathrm{DB}$ intersects at $\mathrm{P},$ then:

In an equilateral triangle $\mathrm{ABC},$ the side $\mathrm{BC}$ is trisected at $\mathrm{D}.$ Find the value of ${\mathrm{AD}}^2 :$

In the figure $∆\mathrm{ACB}~∆\mathrm{APQ}.$ If $\mathrm{BC}=8 \mathrm{cm},\mathrm{PQ}=4 \mathrm{cm},$ $\mathrm{AP}=2.8 \mathrm{cm},$ find $\mathrm{CA} :$

In the figure $BC\parallel AD$. find the value of $x$.