If $DE\Vert BC,AD=4 \text{cm},BD=7.5 \text{cm},AE=6.4 \text{cm }DE=5 \text{cm}$ then the lengths of A C and B C?

By using Thales theorem

$\therefore \frac{AD}{DB}=\frac{AE}{EC}$

$\frac{4}{7.5}=\frac{6.4}{EC}$

$\mathrm{EC}=\frac{7.5\times 6.4}{4}=12 \mathrm{cm}$

$\mathrm{AC}=\mathrm{AE}+\mathrm{EC}=6.4+12=18.4 \mathrm{cm}$

$Also,in ∆ABC and ∆ADE,$

$\angle \mathrm{A}=\angle \mathrm{D}$

$\therefore \Delta \mathrm{ABC}~\Delta \mathrm{ADE}$

$\therefore \frac{AD}{AB}=\frac{DE}{BC}$

$\frac{4}{7.5+4}=\frac{5}{BC}$

$\mathrm{BC}=\frac{11.5\times 5}{4}=14.37 \mathrm{cm}$