An angle bisector of a triangle divides the opposite side of the triangle into segments 5 cm and 9 cm long. A second side of the triangle is 13.5 cm long. Find all possible lengths for the third side of the triangle. Question content area bottom Part 1 The possible​ length(s) for the third side of the triangle​ is(are)    enter your response here cm.

To solve this problem, we can use 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 as x. According to the Angle Bisector Theorem, we have:

5/9 = 13.5/x

Cross-multiplying gives us:

5x = 9 * 13.5

Solving for x, we get:

x = (9 * 13.5) / 5 x = 24.3 cm

However, the third side of a triangle must be less than the sum of the other two sides and greater than the absolute difference of the other two sides. Therefore, we have:

5 + 9 < x < 5 + 13.5 14 < x < 18.5

And

|5 - 9| < x < |5 - 13.5| 4 < x < 8.5

Combining these inequalities, we find that the possible lengths for the third side of the triangle are between 14 cm and 18.5 cm.

To solve this problem, we can use 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 as x. According to the Angle Bisector Theorem, we have:

5/9 = 13.5/x

Cross-multiplying gives us:

5x = 9 * 13.5

Solving for x, we get:

x = (9 * 13.5) / 5 x = 24.3 cm

However, the third side of a triangle must be less than the sum of the other two sides and greater than the absolute difference of the other two sides. So, we have:

5 + 9 < x < 5 + 13.5 14 < x < 18.5

And

|5 - 9| < x < |5 - 13.5| 4 < x < 8.5

Combining these inequalities, we find that the possible lengths for the third side of the triangle are between 14 cm and 18.5 cm.