Vertical angles of two isoceles triangles are equal. Then corresponding altitudes are in the ratio $4: 9.$ Find the ratio of their areas:

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 $x$.

In figure $\mathrm{AD}$ is the bisector of $\angle \mathrm{BAC}$ If $\mathrm{BD}=2 \mathrm{cm},$ $\mathrm{CD}=3 \mathrm{cm}$ and $\mathrm{AB}=5 \mathrm{cm}.$ Find $\mathrm{AC} :$

In the figure $\mathrm{AB}\Vert \mathrm{QR},$ Find the length of $\mathrm{PB}.$

In the figure $\mathrm{QA}$ and $\mathrm{PB}$ are perpendicular to $\mathrm{AB}.$ If $\mathrm{AO}=10 \mathrm{cm},\mathrm{BO}=6 \mathrm{cm}$ and $\mathrm{PB}=9 \mathrm{cm}.$ Find $\mathrm{AQ}.$

In the given figure $\mathrm{AB}=12 \mathrm{cm},\mathrm{AC}=15 \mathrm{cm}$ and $\mathrm{AD}=6 \mathrm{cm}.\mathrm{BC}\Vert \mathrm{DE},$ find the length of $\mathrm{AE}.$

In figure, $\mathrm{ABC}$ is a right angled triangle, right angled at $\mathrm{B} . \mathrm{AD}$ and $\mathrm{CE}$ are the two medians drawn from $\mathrm{A}$ and $\mathrm{C}$ respectively. If $\mathrm{AC}=5 \mathrm{cm}$ and $\mathrm{AD}=3\sqrt{5}/2 \mathrm{cm},$ find the length of $\mathrm{CE}.$