The problem describes a rectangle $ABCD$ with the following dimensions and relationships:

• Sides: $AB = CD = 34 \, \text{cm}$
• $E$ is a point on side $CD$ such that $CE = 9 \, \text{cm}$ and $ED = 25 \, \text{cm}$ (since $CD = 34 \, \text{cm}$, $9 + 25 = 34 \, \text{cm}$).
• $AE$ is perpendicular to diagonal $EB$.

The goal is to find the length of $BC$.

Solution approach:

1. Understand the rectangle: In any rectangle, opposite sides are equal, and all angles are 90 degrees.

2. Geometrical setup:

   • $E$ is a point on $CD$, and a perpendicular line $AE$ from $A$ meets $EB$.
   • The problem suggests a geometric configuration where triangles or Pythagoras' theorem may come into play, given that $AE \perp EB$.

3. Use of Pythagoras' Theorem:

   • In triangle $AEB$, since $AE$ is perpendicular to $EB$, it forms a right triangle. We can apply the Pythagorean theorem to solve for lengths in this triangle.

4. Breaking down diagonals:

   • Use the diagonal properties of the rectangle to help relate the sides.

Let's solve step-by-step:

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