Question

An angle bisector of a triangle divides the opposite side of the triangle into segments 5 cm and 3 cm long. A second side of the triangle is 7.6 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.
a. 12.7 cm, 4.6 cm
b. 15 cm, 4.6 cm
c. 38 cm, 2 cm
d. 38 cm, 12.7 cm

Answer 1

There is a little-known theorem to solve this problem.

The theorem says that
In a triangle, the angle bisector cuts the opposite side into two segments in the ratio of the respective sides lengths.

See the attached triangles for cases 1 and 2.  Let x be the length of the third side.

Case 1:
Segment 5cm is adjacent to the 7.6cm side, then
x/7.6=3/5  => x=7.6*3/5=4.56 cm

Case 2:
Segment 3cm is adjacent to the 7.6 cm side, then
x/7.6=5/3 => x=7.6*5/3=12.67 cm

The theorem can be proved by considering the sine rule on the adjacent triangles ADC and BDC with the common side CD and equal angles ACD and DCB.