An angle bisector of a triangle divides the opposite side of the triangle into segments 6 cm and 4 cm long. a second side of the triangle is 7.4 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. 44.4 cm, 11.1 cm
b. 44.4 cm, 3.2 cm
c. 11.1 cm, 4.9 cm
d. 24 cm, 4.9 cm

The figure for this problem is attached below. To determine the maximum and the minimum lengths of the third side, we use the law of sines. First, we use it to the upper side of the triangle,

3/sin(alpha) = x/sin (beta)

In the lower triangle,
5 / sin (alpha) = 7.5 / sin(pi - beta)

since beta and pi-beta are supplementary angles, then they would have the same sine value. So,
5 / sin (alpha) = 7.5 / sin(pi - beta)
5 / sin (alpha) = 7.5 / sin( beta)

We can now solve for the value of x by dividing the first equation and the equation 2.

3/5 = x/7.5
5x = 3 (7.5)
x = 4.5

We determine the other value of x using the same steps but instead assume the value of 7.4 is on the other side of the figure. With this, the value of x would be 12.5.