The problem asks us to find the value of $x$ in the given right triangle, where the hypotenuse is 12 cm and one of the angles is 45 degrees. The side opposite to the 45-degree angle is $x$.

1. Problem Description

The problem asks us to find the value of $x$ in the given right triangle, where the hypotenuse is 12 cm and one of the angles is 45 degrees. The side opposite to the 45-degree angle is $x$.

2. Solution Steps

Since we have a right triangle with one angle equal to 45 degrees, the other angle must also be 45 degrees (because the sum of angles in a triangle is 180 degrees, and 180 - 90 - 45 = 45). Thus, we have an isosceles right triangle, which means that the side opposite to the 45-degree angle is equal to the side adjacent to the 45-degree angle.

Let the adjacent side also be $x$. Then we have a right triangle with sides $x$, $x$, and a hypotenuse of 12 cm.

By the Pythagorean theorem, we have

$x^2 + x^2 = 12^2$

$2x^2 = 144$

$x^2 = 72$

$x = \sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$

To express $x$ as a decimal, we know $\sqrt{2} \approx 1.414$.

$x = 6 \times 1.414 = 8.484$

We can also use trigonometry: $\sin(45) = \frac{x}{12}$. We know $\sin(45) = \frac{\sqrt{2}}{2}$, so

$x = 12 \sin(45) = 12 \times \frac{\sqrt{2}}{2} = 6\sqrt{2}$

3. Final Answer

$6\sqrt{2}$