A rectangle $A B C D$ has sides $A B=C D=34 \mathrm{~cm} . E$ is a point on $C D$ such that $C E=9 \mathrm{~cm}$ and $E D=25 \mathrm{~cm} . A E$ is perpendicular to $E B$. What is the length of $B C$ ?

Step 1 ：Given a rectangle $A B C D$ with sides $A B=C D=34 \mathrm{~cm}$.

Step 2 ：$E$ is a point on $C D$ such that $C E=9 \mathrm{~cm}$ and $E D=25 \mathrm{~cm}$.

Step 3 ：$A E$ is perpendicular to $E B$.

Step 4 ：We have a right triangle AEB. AE is the height of the rectangle and is equal to BC.

Step 5 ：We know that AB = 34 cm and EB = 34 - 9 = 25 cm.

Step 6 ：We can use the Pythagorean theorem to find the length of BC. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. We can write this rule as $a^2 + b^2 = c^2$ where c represents the length of the hypotenuse and a and b represent the lengths of the other two sides.

Step 7 ：Substituting the given values into the Pythagorean theorem, we get $BC = \sqrt{AB^2 - EB^2} = \sqrt{34^2 - 25^2} = 23.04$ cm.

Step 8 ：Final Answer: The length of $B C$ is \(\boxed{23.04}\) cm.