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
a. 12.7 cm, 4.6 cm
b. 15 cm, 4.6 cm
c. 38 cm, 2 cm
d. 38 cm, 12.7 cm
1 answer:
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.
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.
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