An angle bisector of a triangle divides the opposite side of the triangle into segments 6 cm and 8 cm longA second side of the triangle measures 7 cm long. Find the longest and shortest possible lengths of the third side of the triangle. Round answers to the nearest tenth.
To find the longest and shortest possible lengths of the third side of the triangle, we can apply the triangle inequality theorem which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let x represent the length of the third side. According to the information given, the second side is 7 cm long. Since the angle bisector divides the opposite side into segments 6 cm and 8 cm, we know that the ratio of the lengths of the segments is equal to the ratio of the lengths of the corresponding sides of the triangle. Therefore, we can set up the following proportions:
6/x = 7/8
8/x = 7/6
Now we can solve for x in each proportion:
6/x = 7/8
6(8) = 7x
48 = 7x
x = 48/7
x ≈ 6.9
8/x = 7/6
8(6) = 7x
48 = 7x
x = 48/7
x ≈ 6.9
Therefore, the longest possible length of the third side is approximately 6.9 cm and the shortest possible length is also approximately 6.9 cm.