Finding the length of $AE$ In the figure there're $28$ points .The distance between each point is $1$ unit .The segment $AB$ intersect with the segment $CD$ in the point $E$. How to find the length of $AE$



$$\bigtriangleup AEF \sim\bigtriangleup BED $$ 
This implies 
 $$  \frac{EB}{AE} = \frac{BD}{AF} $$
 $$  \frac{EB}{AE} = \frac{4}{5} $$  
Add $1$ to both sides 
$$ \frac{AB}{AE} = \frac{9}{5} $$
$$ AE = \frac{5}{9} \times AB $$
$$ AE = \frac{5}{9} \times \sqrt{3^2 + 6^2 } = \frac{5}{9} \times \sqrt{45} $$
$$ AE = \frac{5}{9} \times 3\sqrt{5} $$
$$ \boxed{AE = \displaystyle\frac{5\sqrt{5}}{3}} $$
