Basic similarity of triangles problem

Question

The question is to find the length of AB. I'm currently studying civil engineering, but I'm trying to refresh some old knowledge as I got a part time job this semester as a math assistant at a middle school. You're not supposed to use trigonometry.

My attempt involves similarity of triangles of course, since all the angles are obviously identical.

$\frac{AB}{AD} = \frac{3}{6}$

$\frac{AB}{AB+BD} = \frac{1}{2}$

In the solution they somehow know that the length of BD is 4. I figured out that they're at least equal using trigonometry, but I don't see where the 4 comes from.

Answer 1

Updated answer:

The picture in the question is missing $BD = 4$.

From $\frac{AB}{AB+4} = \frac{1}{2}$, we have $2 AB = AB + 4$, so $AB=4$.

Answer 2

There may be more information here than the problem requires.

Since they are right triangles, you know by Pythagoras that$$AB^2+BC^2=AC^2$$ And since $BC= 3$, then if we grant that $5^2$ and $4^2$ are the only two square numbers that differ by $3^2$, it will follow that $AC=5$ and $AB=4$. No trigonometry needed, nor similar triangles.

On the other hand, allowing for irrational numbers, supposing $AC=\sqrt{24}$, $\sqrt{23}$, or $\sqrt{22}$, then by Pythagoras $AB=\sqrt{15}$, $\sqrt{14}$, or $\sqrt{13}$, respectively. We know in every case that $AB=BD$, but beyond that? The problem seems indeterminate, and again similarity of triangles is no help.