$\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$. If $AB + BC = 7$,$XY + YZ = 10.5$,and $AC = 5$,then $XZ = \ldots$

With the help of the definition of the similarity of triangles,prove that all equilateral triangles are similar.

If $AD$ and $PM$ are medians of triangles $ABC$ and $PQR,$ respectively,where $\Delta ABC \sim \Delta PQR,$ prove that $\frac{AB}{PQ} = \frac{AD}{PM}.$

In the figure,if $\angle A = \angle C$,$AB = 6 \, cm$,$BP = 15 \, cm$,$AP = 12 \, cm$,and $CP = 4 \, cm$,then find the lengths of $PD$ and $CD$ (in $cm$).

In the figure,$E$ is a point on the side $CB$ produced of an isosceles triangle $ABC$ with $AB = AC$. If $AD \perp BC$ and $EF \perp AC$,prove that $\Delta ABD \sim \Delta ECF$.

In the figure,two chords $AB$ and $CD$ intersect each other at the point $P$. Prove that $AP \cdot PB = CP \cdot DP$.