We have a right triangle with a hypotenuse of 12 cm and one angle of 45 degrees. We need to find the lengths of the other two sides, labeled as $x$ and $y$.

1. Problem Description

We have a right triangle with a hypotenuse of 12 cm and one angle of 45 degrees. We need to find the lengths of the other two sides, labeled as $x$ and $y$.

2. Solution Steps

Since the triangle is a right triangle and one angle is 45 degrees, the other acute angle must also be 45 degrees (since the sum of angles in a triangle is 180 degrees, and 180 - 90 - 45 = 45). This means the triangle is an isosceles right triangle, and the two legs are equal in length. Therefore, $x = y$.

We can use the sine or cosine function to find the lengths of the sides. Let's use the sine function:

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

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

Since $\sin(45^\circ) = \frac{\sqrt{2}}{2}$, we have:

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

$x = 12 \cdot \frac{\sqrt{2}}{2} = 6\sqrt{2}$

Since $x = y$, we have $y = 6\sqrt{2}$.

3. Final Answer

$x = 6\sqrt{2}$

$y = 6\sqrt{2}$