Question 1210474
The problem as stated is **unsolvable** because a necessary dimension is missing. The length of the side $BC$ (which is equal to $AD$) cannot be uniquely determined without knowing the length of the adjacent side **$CD$** (or $AB$).

Here is the general geometric relationship that must hold in this type of folding problem, which shows the dependency:

Let:
* $BC = AD = l$ (The length you are trying to find)
* $CD = w$ (The missing width)

### 1. Relationships from the Fold

When corner $A$ is folded over the crease $DE$ to point $F$ on $BC$, two triangles, $\triangle ADE$ and $\triangle FDE$, are congruent. This gives us the following equivalences:

* The hypotenuse of the fold: $DF = AD = l$
* The fold sides: $AE = FE$
* $\angle DFE = \angle DAE = 90^\circ$

### 2. Using the Pythagorean Theorem

We can set up two equations using the Pythagorean theorem in the right-angled triangles $\triangle FDC$ and $\triangle FCE$:

* **In $\triangle FDC$ (Right-angled at C):**
    $$DF^2 = DC^2 + CF^2$$
    $$l^2 = w^2 + CF^2 \quad \rightarrow \quad CF = \sqrt{l^2 - w^2}$$

* **In $\triangle FCE$ (Right-angled at C):**
    $$FE^2 = CE^2 + CF^2$$
    Since $FE = AE$, we substitute $AE$ and the expression for $CF^2$:
    $$AE^2 = CE^2 + (l^2 - w^2)$$

### 3. Using the Side $CD$

The line segments $DE$ and $CE$ make up the side $CD$: $CD = DE + CE$.

* **In $\triangle ADE$ (Right-angled at D):**
    $$AE^2 = AD^2 + DE^2$$
    $$AE^2 = l^2 + DE^2$$

### Conclusion of Dependency

By setting the two expressions for $AE^2$ equal to each other:

$$CE^2 + l^2 - w^2 = l^2 + DE^2$$
$$\rightarrow CE^2 = DE^2 + w^2$$

Since $w$ and $l$ are not given, $BC$ (which is $l$) cannot be uniquely determined.

**To solve the problem, you must know the length of $CD$ (or a relationship between $l$ and $w$).**