An angle bisector of a triangle divides the opposite side of the triangle into segments 6 cm and 5 cm long. a second side of the triangle is 6.9 cm long. find the longest and shortest possible lengths of the third side of the triangle. round answers to the nearest tenth of a centimeter.

Based in the information given in the problem, you must apply the The Angle Bisector Theorem. Let's call the triangle: "ABC"; the internal bisector of the angle that divides its opposite side: "AP"; and "x": the longest and shortest possible lengths of the third side of the triangle.

 If BP= 6 cm and CP= 5 cm, we have:

 BP/CP = AB/AC

 We don't know if second side of the triangle (6.9 centimeters long) is AB or AC, so:

 1. If AB = 6.9 cm and AC = x:
 6/5 = 6.9/x
 x = (5x6.9)/6
 x = 5.80 cm

 2. If AC= 6.9 cm and AB= x:
 6/5 = x/6.9
 x = 6.9x6/5
 x = 8.30 cm

 Then, the answer is: 
 The longest possible length of the third side of the triangle is 8.30 cm and the and shortest length of it is 5.80 cm.