To find the length of $BD$ in the given triangle $ABC$, with $BC = 5.7$, $AC = 5.1$, and $CD = 3.2$, we can apply the cosine rule, assuming $\angle BCD$ and $\angle BDC$ are known or can be calculated using the triangle sides.

Step 1: Apply the Cosine Rule in Triangle BCD

We first calculate $\angle BCD$ using the cosine rule in triangle $ACD$:

$\cos(\angle ACD) = \frac{AC^2 + CD^2 - AD^2}{2 \times AC \times CD}$

Given $AC = 5.1$, $CD = 3.2$, and $AD = BD$, we then use the cosine rule to find $BD$:

$BD^2 = BC^2 + CD^2 - 2 \times BC \times CD \times \cos(\angle BCD)$

Step 2: Calculate $BD$

Given $BC = 5.7$ and $CD = 3.2$, the above formula will yield $BD$. Let's compute this step-by-step.

Let's first calculate $\angle BCD$ and then find $BD$.