Question

an angle bisector of a triangle divides the opposite side of the triangle into segments 5 cm and 9 cm long. a second side of the triangle is 13.5 cm long. find all possible lengths for the third side of the triangle.
the possible length(s) for the third side of the triangle is(are) □ cm.
(rounded to one decimal place is needed. use a comma to separate answers as needed.)

Explanation:

Step1: Apply Angle Bisector Theorem

The Angle Bisector Theorem states that the angle bisector divides the opposite side into segments proportional to the adjacent sides. Let the third side length be $x$. We have two cases:
Case 1: $\frac{5}{9} = \frac{13.5}{x}$
Case 2: $\frac{5}{9} = \frac{x}{13.5}$

Step2: Solve Case 1

Rearrange to solve for $x$:
$x = \frac{13.5 \times 9}{5}$
$x = \frac{121.5}{5} = 24.3$

Step3: Solve Case 2

Rearrange to solve for $x$:
$x = \frac{13.5 \times 5}{9}$
$x = \frac{67.5}{9} = 7.5$

Step4: Verify triangle inequality

For $x=24.3$: $5+9=14 < 24.3$? No, wait correction: Wait, the segments are 5 and 9, so the full side is $5+9=14$. The other sides are 13.5 and $x$.
Check triangle inequalities:
For $x=24.3$: $13.5 + 14 > 24.3$ ($27.5>24.3$), $13.5+24.3>14$, $14+24.3>13.5$: valid.
For $x=7.5$: $7.5+13.5>14$ ($21>14$), $7.5+14>13.5$, $13.5+14>7.5$: valid.

Answer:

7.5, 24.3