To find the length of $EB$ in this triangle, we can apply the concept of similar triangles. Since $DE$ is parallel to $AB$, triangles $\triangle CDE$ and $\triangle CAB$ are similar by the Basic Proportionality Theorem (also known as Thales' theorem).

Given:

• $CD = 3.1$
• $CE = 2.5$
• $DA = 1.9$

Let $EB = x$, which we need to find.

Step-by-Step Solution

1. Since $\triangle CDE \sim \triangle CAB$, the ratios of corresponding sides are equal: $\frac{CD}{CA} = \frac{CE}{CB}$

2. First, we find $CA$: $CA = CD + DA = 3.1 + 1.9 = 5.0$

3. Next, we use the proportion: $\frac{CD}{CA} = \frac{CE}{CB}$ Substituting the known values: $\frac{3.1}{5.0} = \frac{2.5}{2.5 + x}$

4. Cross-multiplying to solve for $x$: $3.1 \cdot (2.5 + x) = 5.0 \cdot 2.5$ $7.75 + 3.1x = 12.5$ $3.1x = 4.75$ $x = \frac{4.75}{3.1} \approx 1.5$

Answer

The length of $EB$ is approximately $1.5$ units.

Would you like further details on this process, or do you have any questions?

Here are five related questions to expand your understanding:

1. How would the answer change if $DE$ was not parallel to $AB$?
2. What is the significance of the Basic Proportionality Theorem in similar triangles?
3. How can we apply this technique to solve problems involving parallel lines and transversals?
4. Can this method be used to find unknown side lengths in non-triangular polygons?
5. How would the calculation differ if only the angle measures were given instead of side lengths?

Tip: When working with similar triangles, always identify parallel lines and proportional sides to set up accurate ratios.