We are given a right triangle with hypotenuse of 12 cm and one angle of 45 degrees. We are asked to find the lengths of the other two sides, $x$ (opposite) and $y$ (adjacent).

1. Problem Description

We are given a right triangle with hypotenuse of 12 cm and one angle of 45 degrees. We are asked to find the lengths of the other two sides, $x$ (opposite) and $y$ (adjacent).

2. Solution Steps

We are given that the angle is 45 degrees, the hypotenuse is 12 cm.

We can use the sine function to find the length of the opposite side, $x$:

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

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

$x = 12 * sin(45^\circ)$

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

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

$x = 6\sqrt{2}$

We can use the cosine function to find the length of the adjacent side, $y$:

$cos(\theta) = \frac{adjacent}{hypotenuse}$

$cos(45^\circ) = \frac{y}{12}$

$y = 12 * cos(45^\circ)$

Since $cos(45^\circ) = \frac{\sqrt{2}}{2}$,

$y = 12 * \frac{\sqrt{2}}{2}$

$y = 6\sqrt{2}$

3. Final Answer

$x = 6\sqrt{2}$

$y = 6\sqrt{2}$