We are given a right triangle $\triangle ABC$ with $\angle A = 90^\circ$, $\angle B = 45^\circ$, and $BC = 15\sqrt{2}$. We are tasked with finding $AB$.

Since $\angle B = 45^\circ$ and the sum of angles in a triangle is $180^\circ$, the remaining angle $\angle C$ must also be $45^\circ$. This tells us that $\triangle ABC$ is an isosceles right triangle, where the two legs of the triangle are equal in length.

Step 1: Properties of Isosceles Right Triangle

In an isosceles right triangle (also known as a 45-45-90 triangle), the legs are equal, and the hypotenuse is $\sqrt{2}$ times the length of each leg. Therefore, if the hypotenuse is given as $BC = 15\sqrt{2}$, we can use the following formula for a 45-45-90 triangle:

$BC = AB \times \sqrt{2}$

Step 2: Solving for $AB$

We can rearrange the equation to solve for $AB$:

$AB = \frac{BC}{\sqrt{2}} = \frac{15\sqrt{2}}{\sqrt{2}} = 15$

Thus, the length of $AB$ is $15$.

Final Answer:

$AB = 15$

Would you like more details or further explanation?

Here are 5 related questions for further exploration:

1. How do you derive the properties of a 45-45-90 triangle?
2. What is the length of $AC$ in this triangle?
3. Can you solve for the area of $\triangle ABC$?
4. What is the perimeter of $\triangle ABC$?
5. How do the properties of other special triangles (e.g., 30-60-90 triangles) differ from 45-45-90 triangles?

Tip: In any right triangle, the sum of the two acute angles is always $90^\circ$.