The problem is to find the length of the side $x$ opposite to the $45^{\circ}$ angle in a right triangle, given that the hypotenuse is $12$ cm.

1. Problem Description

The problem is to find the length of the side $x$ opposite to the $45^{\circ}$ angle in a right triangle, given that the hypotenuse is $12$ cm.

2. Solution Steps

We can use the sine function to relate the opposite side, hypotenuse, and the angle. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.

$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

In our case, the angle $\theta = 45^{\circ}$, the opposite side is $x$, and the hypotenuse is $12$ cm. Therefore, we have:

$\sin(45^{\circ}) = \frac{x}{12}$

We know that $\sin(45^{\circ}) = \frac{\sqrt{2}}{2}$. So, we can write:

$\frac{\sqrt{2}}{2} = \frac{x}{12}$

To find $x$, we can multiply both sides of the equation by 12:

$x = 12 \cdot \frac{\sqrt{2}}{2}$

$x = 6\sqrt{2}$

3. Final Answer

$x = 6\sqrt{2}$