Given that ABC = 90^, ABC is a right-angled triangle.

Given that $\angle ABC = 90^\circ$, $\triangle ABC$ is a right-angled triangle.
The lengths of the two shorter sides are $AB = 4$ cm and $BC = 3$ cm.

In $\triangle ABC$:

The area of a right-angled triangle can be calculated as half the product of its perpendicular sides.

Using AB and BC as base and height:

Given that $\angle BDC = 90^\circ$, BD is the altitude from B to the hypotenuse AC.
We can also express the area of $\triangle ABC$ using AC as the base and BD as the height:

We know the Area is $6\, \text{cm}^2$ and $AC = 5\, \text{cm}$.

Multiply both sides by 2:

Divide by $5\, \text{cm}$:

The length of BD is $\text{2.4 cm}$.