We are given a right triangle with a hypotenuse of 12 cm and one angle measuring 45 degrees. We need to find the length of the side opposite the 45-degree angle, which is labeled as $x$.

1. Problem Description

We are given a right triangle with a hypotenuse of 12 cm and one angle measuring 45 degrees. We need to find the length of the side opposite the 45-degree angle, which is labeled as $x$.

2. Solution Steps

Since we have a 45-degree angle in a right triangle, we can use trigonometric ratios (SOH CAH TOA) to find the unknown side. We are given the hypotenuse (hyp) and we need to find the opposite side (opp). The sine function relates the opposite side to the hypotenuse:

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

In our case, $\theta = 45$ degrees, the hypotenuse is 12 cm, and the opposite side is $x$. So we have:

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

Now, we need to solve for $x$. We know that $sin(45) = \frac{\sqrt{2}}{2}$.

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

Multiply both sides by 12 to isolate $x$:

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

$x = 6\sqrt{2}$

Now, we need to approximate this value as a decimal. Since $\sqrt{2} \approx 1.414$, we have:

$x \approx 6 \cdot 1.414$

$x \approx 8.484$

3. Final Answer

$x = 6\sqrt{2} \approx 8.484$