To solve this problem, we will use the Angle Bisector Theorem, which states that when an angle bisector divides an angle, it divides the opposite side in the same ratio as the adjacent sides.

For triangle ABC, the segment BD bisects angle ABC, so the Angle Bisector Theorem gives the equation:

$\frac{AD}{DB} = \frac{AC}{BC}$

From the image:

• $AC = 14$
• $BC = 17$
• $AD = x$
• $DB = 13$

Now we can substitute the known values into the ratio:

$\frac{x}{13} = \frac{14}{17}$

To solve for $x$, we cross-multiply:

$x \times 17 = 14 \times 13$

$17x = 182$

$x = \frac{182}{17} \approx 10.7$

Thus, the value of $x$ is approximately 10.7.

Would you like further details on the solution, or do you have any questions?

Here are 5 related questions you might find helpful:

1. How does the Angle Bisector Theorem help in triangle geometry?
2. How can you apply the Angle Bisector Theorem in other types of triangles?
3. What happens if the angle bisector is not drawn from the vertex?
4. How do you find unknown side lengths when given ratios of sides in triangle problems?
5. Can the Angle Bisector Theorem be applied to non-right triangles?

Tip: When using the Angle Bisector Theorem, always make sure the ratio of the sides is set up correctly before solving.