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
A. 7.4 cm and 9.3 cm
B. 8.3 cm and 5.8 cm
C. 4.2 cm and 7.6 cm
D. 5.3 cm and 9.3 cm
To solve this problem, we can use the Angle Bisector Theorem.
According to the Angle Bisector Theorem, the ratio of the lengths of the segments on the opposite side of the triangle created by the angle bisector is equal to the ratio of the lengths of the other two sides of the triangle.
Let the third side of the triangle be x cm. Then, we can set up the following proportion:
6/8 = 7/x
Cross multiplying, we have: 6x = 8 * 7
Simplifying, we get: 6x = 56
Dividing both sides by 6, we find: x = 56/6
Simplifying further, we get: x = 9.3
Therefore, the longest possible length of the third side of the triangle is 9.3 cm.
For the shortest possible length of the third side, we need to consider the extreme case where the angle bisector coincides with the side of length 7 cm. In this case, the third side would have a length of 7 - 6 = 1 cm.
Therefore, the shortest possible length of the third side of the triangle is 1 cm.
Rounding to the nearest tenth, the answer is (B) 1 cm and 9.3 cm.