The problem presents a right triangle with one angle of 45 degrees. The hypotenuse is labeled as 12 cm. We are asked to find the lengths of the other two sides, labeled as $x$ and $y$.

1. Problem Description

The problem presents a right triangle with one angle of 45 degrees. The hypotenuse is labeled as 12 cm. We are asked to find the lengths of the other two sides, labeled as $x$ and $y$.

2. Solution Steps

Since the triangle is a right triangle with one angle of 45 degrees, the other acute angle must also be 45 degrees (because the angles in a triangle add up to 180 degrees, and $180 - 90 - 45 = 45$). This means that the triangle is a 45-45-90 triangle, and therefore, it is an isosceles right triangle, with legs of equal length. So, $x = y$.

In a 45-45-90 triangle, the ratio of the lengths of the sides is $1:1:\sqrt{2}$. If the length of each leg is $s$, then the length of the hypotenuse is $s\sqrt{2}$. In this problem, the length of the hypotenuse is 12 cm. Therefore, we have $s\sqrt{2} = 12$.

To solve for $s$, we divide both sides by $\sqrt{2}$:

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

We can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$:

$s = \frac{12\sqrt{2}}{2} = 6\sqrt{2}$

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

3. Final Answer

$x = 6\sqrt{2}$

$y = 6\sqrt{2}$