In triangle ABC, let the angle bisector of angle A intersect side BC at point D, dividing BC into segments BD = 6 cm and DC = 5 cm. According to the angle bisector theorem, the ratio of the two segments on the opposite side is equal to the ratio of the other two sides of the triangle. Hence, we can write:
\[ \frac{AB}{AC} = \frac{BD}{DC} = \frac{6}{5} \]
Let the length of side AC be represented as 5x and the length of side AB be represented as 6x, where x is a common multiplier. The total length of side AC is given as 69 cm. Therefore, we can set up the equation:
\[ 5x + 6x = 69 \]
Combining like terms, we obtain:
\[ 11x = 69 \]
Solving for x gives:
\[ x = \frac{69}{11} = 6.27 \]
Now, we can compute the lengths of sides AB and AC:
\[ AB = 6x = 6 \times 6.27 = 37.62 \text{ cm} \]
\[ AC = 5x = 5 \times 6.27 = 31.35 \text{ cm} \]
Thus, we have determined the lengths of sides AB and AC using the properties of the angle bisector and the given information about the triangle. Side AB measures approximately 37.62 cm, while side AC measures approximately 31.35 cm.