Final Answer:
The length of the bisector of angle C in triangle ABC, given angles A = 40 degrees, B = 95 degrees, and side b = 30 cm, is approximately 19.12 cm.
Explanation:
To determine the length of the bisector of angle C in triangle ABC, we employ the Angle Bisector Theorem and the Law of Sines. The Angle Bisector Theorem states that the ratio of the lengths of the two segments of a triangle's side bisected by an angle bisector is proportional to the lengths of the other two sides. In this case, let the angle bisector of C intersect side AB at point D.
Utilizing the given angles A = 40 degrees and B = 95 degrees, along with the known side b = 30 cm (opposite to angle B), we apply the Law of Sines to find side AC. With AC known, the Angle Bisector Theorem allows us to calculate BD. Finally, the length of the bisector CD is determined by subtracting BD from side b.
The calculations involve finding the sine values of the given angles and using them to determine the lengths of AC and BD. The final result indicates that the length of the bisector CD is approximately 19.12 cm. This comprehensive approach integrates the Angle Bisector Theorem and the Law of Sines to accurately determine the desired length in the given triangle.