Length of Triangle BCD 
Hey, well I'm doing some higher level revision and I'm stuck... 
In the diagram triangle BCD is mathematically similar to triangle ACE.
So what is the length of BD? How do you work it out? 
 A: You can easily find the length of DB from $\triangle ACE\sim \triangle BCD$, the coefficient of similarity is evident from the side $CE$. On the other hand, you can't recover the length $BC$ unless you have any additional information. You may draw several triangles satisfying those lengths but with different $\angle CEA$ (and therefore, different lengths of $AC$).
A: $$4:DB=10:7\frac{1}{2}\Rightarrow DB=\frac{7\frac{1}{2}\cdot 4}{10}=3$$
A: The ratio DB/EA must equal the ratio DC/EC. Take it from there.
A: Your problem tells you that the two triangles are similar triangles. Similar triangles always have the same three angles (even if they are rotated).
When we have two sides, the angle between them determines the length of the third side, the side opposite the angle. Because the angles of the two triangles are the same (because they are similar triangles), then the sides must be proportionate to one another. 
Because the sides are proportionate to one another, we can work out ratios: let's take one side from triangle A and one side from triangle B, both of which are between the same two angles (or opposite the same angle). Given that we then have the ratio, we can work out the length of the side we are trying to find.
So let's take side DC $=4$ and EC $=(4+6)=10.$ Let's divide the smaller side by the bigger side: $\frac{DC}{EC} = \frac{4}{10}.$ Let's call $\frac{4}{10}$ the ratio. 
Let's look at the ratio a bit more:
$\frac{Smaller Side}{Bigger Side}= ratio$
 then $Smaller Side = ratio \cdot BiggerSide$
 and $\frac{SmallerSide}{ratio} = BiggerSide$
Now that we have this, we can use it to work out side BD. In this case, BD is the smaller side, and EA is the bigger side. So we use the ratio and the bigger side to find the smaller side:  $Smaller Side = ratio \cdot BiggerSide$
  i.e. $BD = ratio \cdot EA$
$BD = \frac{4}{10} \cdot 7.5$
$\therefore BD = 3$
