Area Comparison of Triangles in a Divided Triangle

Consider triangle ABC, where point D is on side BC such that AD is the angle bisector of angle A. If the lengths of sides AB and AC are 12 cm and 16 cm respectively, and the length of side BC is 20 cm, determine the differences in the areas of triangle ABD and triangle ADC.

Use the formula for the area of a triangle given by:

$$ ext{Area} =\frac{1}{2} imes ext{base} imes ext{height}$$

Keep in mind that the area of triangle ABC can be calculated using Heron's formula:

$$s =\frac{a + b + c}{2}$$

where $s$ is the semi-perimeter, and $a$, $b$, and $c$ are the lengths of the sides of the triangle.