We have a right triangle with a hypotenuse of 12 cm and one angle measuring 45 degrees. We need to find the lengths of the other two sides, $x$ (opposite) and $y$ (adjacent), rounded to three decimal places.

1. Problem Description

We have a right triangle with a hypotenuse of 12 cm and one angle measuring 45 degrees. We need to find the lengths of the other two sides, $x$ (opposite) and $y$ (adjacent), rounded to three decimal places.

2. Solution Steps

Since we have a 45-degree angle in a right triangle, this is a 45-45-90 triangle. However, we can also use trigonometric ratios.

First, let's find $x$, which is opposite to the 45-degree angle. We can use the sine function:

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

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

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

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

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

Now, let's find $y$, which is adjacent to the 45-degree angle. We can use the cosine function:

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

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

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

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

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

Alternatively, since it is a 45-45-90 triangle, the side opposite to the 45-degree angle is equal to the adjacent side. Therefore $x = y$.

$x = 6\sqrt{2} \approx 6 \cdot 1.41421356 = 8.48528137$

$y = 6\sqrt{2} \approx 6 \cdot 1.41421356 = 8.48528137$

Rounding to three decimal places:

$x \approx 8.485$

$y \approx 8.485$

3. Final Answer

$x = 6\sqrt{2} \approx 8.485$

$y = 6\sqrt{2} \approx 8.485$