Final answer:
The Angle Bisector Theorem, which states that the ratio of the lengths of AB to AC is equal to the ratio of lengths of BD to DC, is used to calculate AC as 8 cm. However, this option is not listed among the choices provided, suggesting there may be a typo in the question.
Step-by-step explanation:
To find the length of AC in triangle ABC, where AD is the bisector of angle BAC and intersects BC at point D, we can use the Angle Bisector Theorem. This theorem states that the line segment AD bisects the angle BAC, the ratio of the length of the side AB to the length of the side AC is equal to the ratio of the length of BD to the length of DC.
Given that AB = 8 cm, BC = 6 cm, and DC = 3 cm, we can set up the following ratio:
AB / AC = BD / DC
Since BD + DC = BC, we can find BD as:
BD = BC - DC = 6 cm - 3 cm = 3 cm
Now we can use the ratio to find AC:
8 cm / AC = 3 cm / 3 cm
Multiplying both sides by AC and then by 3 cm, we get:
8 cm × 3 cm = AC × 3 cm
24 cm = 3 × AC
AC = 24 cm / 3
AC = 8 cm
Therefore, the length of AC in the triangle ABC is 8 cm, but since this option is not available, it's likely there is a typo in the question or answer choices. In such cases, it is important to verify the question and check for any discrepancies.