The ratio of the corresponding altitudes of two similar triangles is $\frac{3}{5}$. The ratio of their areas is $\frac{6}{5}$.

$D$ is a point on side $QR$ of $△\mathrm{PQR}$ such that $\mathrm{PD}\perp \mathrm{QR}$. Will it be correct to say that $△\mathrm{PQD}~△\mathrm{RPD}$ ? Why ?

In the figure, if $\angle D=\angle C$, then $△ADE~△ACB$.

If in two triangles, an angle of one triangle is equal to an angle of another triangle and two sides of one triangle are proportional to the two sides of the other triangle, then the triangles are similar.

In a $△\mathrm{PQR},{\mathrm{PR}}^2-{\mathrm{PQ}}^2={\mathrm{QR}}^2$ and $M$ is a point on side $P$$R$ such that $\mathrm{QM}\perp \mathrm{PR}$. Prove that ${\mathrm{QM}}^2=\mathrm{PM}\times \mathrm{MR}$.

Find the value of x for which $DE\left|\right|AB$ in given figure.

In figure, if $\angle 1=\angle 2$ and $△NSQ\cong △MTR$, then prove that $△PTS~△PRQ$.

Diagonals of a trapezium $PQRS$ intersect each other at the point $O$, $PQ\left|\right|RS$ and $PQ=3RS$. Find the ratio of the areas of $△POQ$ and $△ROS$.

In figure, if $AB\left|\right|DC$ and $AC,PQ$ intersect each other at the point $O$. Prove that $OA.CQ=OC.AP$.