The bisector of ∠B in ∆ABC meets AC at D. If AB = 12 cm, BC = 18 cm and AC = 15 cm, then the length of AD (in cm) is:

To find the length of segment AD, we can apply the Angle Bisector Theorem to triangle ABC.

The Angle Bisector Theorem states that the angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle.

In $△ABC$, BD is the bisector of $\angle B$ meeting AC at D. Therefore, according to the theorem:
$\frac{AD}{DC}=\frac{AB}{BC}$

We are given the following lengths:
AB = 12 cm
BC = 18 cm
AC = 15 cm

Let the length of AD be $x$ cm. Since AC = 15 cm, the length of DC will be:
$DC=15-x$

Now, substitute these values into the angle bisector ratio:
$\frac{x}{15-x}=\frac{12}{18}$

Simplify the fraction on the right side by dividing the numerator and the denominator by their greatest common divisor, which is 6:
$\frac{x}{15-x}=\frac{2}{3}$

Cross-multiply to solve for $x$:
$3x=2(15-x)$
$3x=30-2x$

Add $2x$ to both sides of the equation:
$3x+2x=30$
$5x=30$

Divide both sides by 5:
$x=\frac{30}{5}$
$x=6$

Thus, the length of AD is 6 cm.