An angle bisector of a triangle divides the opposite side of the triangle into segments 6 cm and 8 cm long. A 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. (1 point)
• 3.5 cm and 9.3 cm
• 4.2 cm and 7.6 cm
7.4 cm and 9.3 cm
8.3 cm and 5.8 cm
Let's call the length of the third side of the triangle "x".
According to the angle bisector theorem, the ratio of the lengths of the segments of the opposite side is equal to the ratio of the lengths of the sides they intersect.
So, in this case, we have:
6 cm / 8 cm = 7 cm / x
Cross-multiplying and simplifying, we get:
6x = 56
x = 56/6
x = 9.333333...
So, the shortest possible length of the third side is approximately 9.3 cm.
To find the longest possible length of the third side, we can use 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.
So, we have:
7 cm + 8 cm > x
15 cm > x
Therefore, the longest possible length of the third side is 15 cm.
Rounding to the nearest tenth, the answer is:
Shortest length: 9.3 cm
Longest length: 15.0 cm
Therefore, the correct answer is:
• 9.3 cm and 15.0 cm