An angle bisector of a triangle divides the opposite side of the triangle into segments 4 cm and 5 cm long. A second side of the triangle is 12.5 cm long. Find all possible lengths for the third side of the triangle.

This problem can be solved using the Angle Bisector Theorem, which states that the ratio of the lengths of the two segments created by the angle bisector is equal to the ratio of the lengths of the other two sides of the triangle.

Let's denote:

• the length of the third side of the triangle as x,
• the length of the side opposite to the angle bisector as a = 4 cm + 5 cm = 9 cm,
• the length of the second side of the triangle as b = 12.5 cm,
• the lengths of the two segments created by the angle bisector as c = 4 cm and d = 5 cm.

According to the Angle Bisector Theorem, we have:

c/d = a/b 4/5 = 9/12.5

This equation holds true, so the given lengths are possible for a triangle.

Now, to find the possible lengths for the third side of the triangle, we can use the Triangle Inequality Theorem, which states that the length of any side of a triangle is less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides.

So, we have:

|a - b| < x < a + b |9 - 12.5| < x < 9 + 12.5 3.5 < x < 21.5

Therefore, the possible lengths for the third side of the triangle are all values greater than 3.5 cm and less than 21.5 cm.