**Given Information:**

- A triangle ABC is drawn to circumscribe a circle of radius 4cm.
- The point of contact of the circle with the triangle is D.
- The lengths of segments BD and DC are 8cm and 6cm, respectively.

**To Find:**

We need to find the lengths of sides AB and AC.

**Solution:**

Let's first analyze the given information and understand the properties of a circle inscribed in a triangle.

**Properties of a Circle Inscribed in a Triangle:**

1. The point of contact of the circle with the triangle lies on the angle bisectors of the triangle.
2. The lengths of the segments formed by the point of contact on the sides of the triangle are equal to the tangents drawn from any vertex of the triangle to the inscribed circle.
3. The lengths of the tangent segments are given by the formula:

tangent length = $\sqrt{(s-a)(s-b)(s-c)/s}$

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

**Using the Properties to Solve the Problem:**

1. Let's consider the triangle ABC and the circle inscribed in it.

2. We know that the lengths of segments BD and DC are 8cm and 6cm, respectively.

3. Using property 2, we can say that BD and DC are the lengths of tangents drawn from vertices B and C to the inscribed circle.

4. Applying property 3, we can write the following equations:

$\sqrt{(s-a)(s-b)(s-c)/s}$ = 8 ...(1)

$\sqrt{(s-a)(s-b)(s-c)/s}$ = 6 ...(2)

5. Since the radius of the inscribed circle is given as 4cm, we can write:

$\sqrt{(s-a)(s-b)(s-c)/s}$ = 4 ...(3)

6. Solving equations (1), (2), and (3) simultaneously will give us the values of sides AB and AC.

7. Let's assume the lengths of sides AB and AC as x and y, respectively.

8. Using the formula for the semi-perimeter of a triangle:

s = (x + y + z) / 2

where z is the length of side BC.

9. We also know that z = BD + DC = 8 + 6 = 14cm.

10. Substituting the values in equation (3) and simplifying, we get:

$\sqrt{(s-x)(s-y)(s-14)/s}$ = 4

$\sqrt{(x+y+14-x)(x+y+14-y)(x+y+14-14)/(x+y+14)}$ = 4

$\sqrt{xy(x+y+14)/(x+y+14)}$ = 4

$\sqrt{xy}$ = 4

xy = 16

11. We have obtained a new equation xy = 16, which gives us a relationship between the lengths of sides AB and AC.

12. To find the specific values of AB and AC, we need one more equation.

13. We can use the formula for the area of a triangle:

Area = $\sqrt{s(s-a)(s-b)(s-c)}$