Problem 235

We begin with a diagram:

The bases of a trapezoid are parallel by definition, so $\angle EDC$ and $\angle EBA$ are alternate interior angles, and therefore equal. We have the same setup with $\angle ECD$ and $\angle EAB$, meaning that $\triangle ABE \sim \triangle CDE$ by AA Similarity. We could've also used the fact that $\angle BEA$ and $\angle DEC$ are vertical angles.
With this information, we can setup a ratio of corresponding sides:
$$\frac{AB}{CD} = \frac{AE}{CE} \implies \frac{2CD}{CD} = \frac{11 - CE}{CE}.$$
And simplify from there:
$$\begin{align*} \frac{2CD}{CD} &= \frac{11 - CE}{CE} \\ 2 &= \frac{11 - CE}{CE} \\ 2CE &= 11 - CE \\ 3CE &= 11 \\ CE &= \frac{11}{3} = 3 \frac{2}{3}. \end{align*}$$
Therefore, our answer is $\boxed{\textbf{(A) }3\textstyle\frac{2}{3}.}$

1. Identify the given information and draw the trapezoid:
- Let $ABCD$ be a trapezoid with $AB$ as the longer base and $DC$ as the shorter base.
- Given that $AB = 2 \cdot DC$.
- Let $E$ be the point of intersection of the diagonals $AC$ and $BD$.
- The length of diagonal $AC$ is given as $11$.

2. Establish the similarity of triangles:
- Since $ABCD$ is a trapezoid with $AB \parallel DC$, the triangles $ABE$ and $CDE$ are similar by the AA (Angle-Angle) similarity criterion.
- This similarity gives us the ratio of corresponding sides:
$$\frac{AB}{DC} = \frac{AE}{EC}$$
- Given $AB = 2 \cdot DC$, we have:
$$\frac{AB}{DC} = 2$$
- Therefore:
$$\frac{AE}{EC} = 2$$

3. **Express $AE$ in terms of $EC$:**
- Let $EC = x$. Then $AE = 2x$ because $\frac{AE}{EC} = 2$.

4. **Use the length of diagonal $AC$:**
- The total length of $AC$ is given as $11$.
- Therefore:
$$AE + EC = 11$$
- Substituting $AE = 2x$ and $EC = x$:
$$2x + x = 11$$
$$3x = 11$$
$$x = \frac{11}{3}$$

5. **Determine the length of $EC$:**
- Since $x = EC$, we have:
$$EC = \frac{11}{3}$$

The final answer is $\boxed{3 \frac{2}{3}}$