An angle bisector of a triangle divides the opposite side of the triangle into segments 6 cm and 5 cm long. The 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.

Solution:

Given, angle bisector divides the opposite side of the triangle into segments 6 cm and 5 cm long.

Second side of the triangle is 6.9 cm.

We have to find the longest and shortest possible lengths of the third side of the triangle.

We know. Each angle bisector of a triangle divides the opposite side into segments proportional in length to the adjacent sides.

In triangle ABC, AD is the angle bisector of angle A such that D is on the side BC.

Now, BD / DC = AB / AC

BC = 6 + 5 = 11cm

Angle bisector divides BC into 2 segments, 

BD = 6 cm 

CD = 5 cm

If AB = 6.9 cm,

6/5 = 6.9 / AC

AC = 6.9(5) / 6

AC = 32.5 / 6

AC = 5.75 cm

If AC = 6.9 cm

6/5 = AB/6.9

AB = 6(6.9)/5

AB = 41.4/5

AB = 8.28 cm

Therefore, the longest and shortest possible lengths of the third side of the triangle are 8.3 cm and 5.8 cm.

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An angle bisector of a triangle divides the opposite side of the triangle into segments 6 cm and 5 cm long. The 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.

Summary:

An angle bisector of a triangle divides the opposite side of the triangle into segments 6 cm and 5cm long. The second side of the triangle is 6.9 cm long. The longest and shortest possible lengths of the third side of the triangle are 8.3 cm and 5.8 cm.