Area Δ ABC = 1/2 . 4 . (AB + BC + AC) = root( s(s – a)(s – b)(s – c))i.e., 4 s = root( s(s – a)(s – b)(s – c) )16 s = (s – a) (s – b) (s – c) i.e., 16 (14 + x) = x X 6 X 8,   i.e., x = 7Therefore, AB = 15 cm and AC = 13 cm.

Given information:
- The radius of the incircle of a triangle is 4 cm.
- One side of the triangle is divided by the point of contact into two segments, measuring 6 cm and 8 cm.

Approach:
To solve this problem, we can use the properties of the incenter and the incircle of a triangle. The incenter is the point of concurrency of the angle bisectors of a triangle, and the incircle is the circle that is tangent to all three sides of the triangle.

Properties of the Incenter and Incircle:
1. The incenter is equidistant from the three sides of the triangle.
2. The line segment from the incenter to the point of contact on a side is perpendicular to that side.
3. The radius of the incircle is perpendicular to the sides of the triangle and bisects them.

Solution:

Step 1: Drawing the Triangle and Incircle
- Draw a triangle ABC.
- Label the sides as AB, BC, and AC.
- Draw a circle inside the triangle such that it is tangent to all three sides.
- Label the point of contact on AB as D, on BC as E, and on AC as F.

Step 2: Identifying the Incenter
- Draw the angle bisectors of the triangle at angles A, B, and C.
- The point of intersection of these angle bisectors will be the incenter, denoted as I.

Step 3: Applying the Properties of Incenter and Incircle
- Apply property 1: The incenter is equidistant from the three sides of the triangle. Therefore, AI = BI = CI.
- Apply property 2: The line segment from the incenter to the point of contact on a side is perpendicular to that side. Therefore, ID ⊥ AB, IE ⊥ BC, and IF ⊥ AC.
- Apply property 3: The radius of the incircle is perpendicular to the sides of the triangle and bisects them. Therefore, ID = IE = IF = 4 cm.

Step 4: Finding the Lengths of AB, BC, and AC
- Let x be the length of AB and y be the length of AC.
- Since D divides AB into two segments of 6 cm and 8 cm, we can write the equation: x = 6 + 8 = 14 cm.
- Similarly, since F divides AC into two segments of 6 cm and y, we can write the equation: y = 6 + y.
- Note that ID is the perpendicular bisector of AB, so it divides AB into two equal segments of length 7 cm each.
- Using the Pythagorean theorem, we can find the length of IC: IC^2 = ID^2 + DC^2.
- Substituting the values, we get IC^2 = 4^2 + 7^2 = 16 + 49 = 65.
- Since IC = AI = BI, we can conclude that IC = AI = BI = √65 cm.

Answer:
The lengths of the sides of the triangle are:
- AB = 14 cm
- BC = 2(IC) = 2(√65) cm
-