We are dealing with a triangle $ABC$ with an exterior angle $\angle DAB$ at vertex $A$, where $AE$ is the bisector of this exterior angle. We are given the lengths of the sides:

• $AC = 18 \, \text{cm}$
• $AB = 12 \, \text{cm}$
• $BC = 8 \, \text{cm}$

Our task is to find the length of $EB$, where $E$ lies on line segment $AB$ due to the bisector.

Approach:

Since $AE$ is the bisector of the exterior angle $\angle DAB$, we can apply the Angle Bisector Theorem. The theorem states that the ratio of the sides opposite to the two angles formed by the bisector is equal to the ratio of the other two sides of the triangle.

In other words: $\frac{BE}{EA} = \frac{BC}{AC}$

Step-by-step solution:

1. According to the Angle Bisector Theorem: $\frac{BE}{EA} = \frac{BC}{AC} = \frac{8}{18} = \frac{4}{9}$

2. Let $BE = x$ and $EA = 12 - x$, because the total length of $AB = 12 \, \text{cm}$.

3. Using the ratio from the Angle Bisector Theorem: $\frac{x}{12 - x} = \frac{4}{9}$

4. Cross-multiply to solve for $x$: $9x = 4(12 - x)$

5. Expand the equation: $9x = 48 - 4x$

6. Combine like terms: $9x + 4x = 48$ $13x = 48$

7. Solve for $x$: $x = \frac{48}{13} \approx 3.69 \, \text{cm}$

Thus, the length of $EB$ is approximately 3.69 cm.

Would you like any further clarification or details on the steps? Here are some related questions you might want to explore:

1. How does the Angle Bisector Theorem work in different triangle configurations?
2. What if the given triangle is not scalene?
3. How can we apply the exterior angle property in different problems?
4. Could we use the Law of Cosines to solve this problem?
5. How does the Side-Splitter Theorem compare to the Angle Bisector Theorem?

Tip: The Angle Bisector Theorem is very useful for dividing segments in proportion when solving geometry problems.