Calculating Perpendicular and Base of Triangle. Suggestion

Question

In this diagram AB and CD are both perpendicular to BE.If EC=5 and CD=4. What is ratio of AB to BE ?

How would i go about solving this triangle (without trigonometric ratios). I could only get DE=3 using Pythagoras theorem and was stuck after that. How would i calculate BD ? Do i make BC ? Suggestions ?

Edit: The answer is 4:3.

Answer 1

By Thales' theorem we have $\frac{CD}{AB}=\frac{DE}{BE}$ and so $\frac{AB}{BE}=\frac{CD}{DE}$.
We know $CD=4$ and by the Pythagorean theorem in $CDE$ we get $DE=3$.
So $\frac{AB}{BE}=\frac{4}{3}$.

Answer 2

The two triangles $ABE$ and $CDE$ are similar. They have the same shape, You can think of $\triangle ABE$ as $\triangle CDE$, after you have put $\triangle CDE$ into a copier and scaled it up somewhat.

So the ratio of $AB$ to $BE$ is the same as the ratio of $CD$ to $DE$. But you calculated $DE$, so you know this ratio.