In $\Delta ABC$ and $\Delta XYZ$,$\angle A \cong \angle Y$ and $\angle B \cong \angle Z$. If $\frac{AC}{YX} = \frac{5}{7}$ and $AB = 7$,find $YZ$.

In $\Delta ABC$,$A-N-B$,$A-M-C$ and $B-X-C$. $\overline{XM} \parallel \overline{AB}$ and $\overline{XN} \parallel \overline{AC}$. $\overline{MN}$ intersects $\overline{CB}$ at $T$. Prove that $TX^2 = TB \times TC$.

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 $cm$).

$\Delta XYZ \sim \Delta DEF$ for the correspondence $XYZ \leftrightarrow EFD$. If $XY : EF = 2 : 3$ and the area of $\Delta XYZ$ is $18$,then the area of $\Delta DEF$ is........

In $\Delta DEF, m \angle D = 90^{\circ}$. If $EF = 6$ and $DF = 4$,then $DE = \dots$

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = 6$,$MB = 12$ and $AN = 8$,find $AC$.