This triangle is a right triangle with a 45° angle, meaning it is a 45°-45°-90° triangle. The sides of a 45°-45°-90° triangle are in a fixed ratio:

$1 : 1 : \sqrt{2}$

The hypotenuse is $\sqrt{2}$ times the length of either leg. Here, the hypotenuse is given as $3\sqrt{2}$, and we need to find $x$, one of the legs.

Step-by-Step Solution:

1. The relationship between the hypotenuse and a leg in a $45^\circ - 45^\circ - 90^\circ$ triangle is: $\text{Hypotenuse} = \text{Leg} \cdot \sqrt{2}$

2. Substituting the hypotenuse $3\sqrt{2}$: $3\sqrt{2} = x \cdot \sqrt{2}$

3. Divide both sides by $\sqrt{2}$: $x = 3$

Final Answer:

The length of $x$ is: $\boxed{3}$

Would you like a more detailed explanation or any clarification?

Related Questions:

1. What are the properties of a $45^\circ - 45^\circ - 90^\circ$ triangle?
2. How does the hypotenuse-to-leg ratio differ in other right triangles (e.g., $30^\circ - 60^\circ - 90^\circ$)?
3. What happens if the given hypotenuse were a non-radical number? How would you solve it?
4. Can the Pythagorean theorem be used to confirm the solution here?
5. How can this problem be extended to coordinate geometry?

Tip:

In a $45^\circ - 45^\circ - 90^\circ$ triangle, if one side is known, all other sides can be calculated quickly using the $1:1:\sqrt{2}$ ratio!