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 $cm$).

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

In $\square ABCD$,$\overline{AB} \parallel \overline{CD}$ and $\overline{AC} \cap \overline{BD} = \{M\}$. If $\frac{MB}{MD} = \frac{5}{3}$ and $CD = 12$,find $AB$.

In $\Delta ABC$ and $\Delta PQR$,$\angle A \cong \angle R$ and $\angle B \cong \angle P$. If $AB = 6$,$BC = 8$ and $PR = 9$,then $PQ = \ldots$

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = x-3$,$MB = 2x-7$,$AN = x+3$ and $NC = 2x+3$,find the value of $x$.

When a ladder leans on a wall to reach a height of $8 \, m$, its lower end rests $6 \, m$ away from the base of the wall. If the same ladder leans on the wall to reach a height of $6 \, m$, find the distance of its lower end from the base of the wall. (in $m$)