The image shows a right triangle inscribed in a circle. The right angle of the triangle touches the circumference, and there is an angle labeled $52^\circ$. One leg of the triangle is given as $6 \text{ cm}$, and the other leg is labeled $x \text{ cm}$.

To solve for $x$, we will use trigonometric ratios. Specifically, we can use the tangent function, which relates the opposite and adjacent sides of a right triangle:

$\tan(52^\circ) = \frac{6}{x}$

Solving for $x$, we get:

$x = \frac{6}{\tan(52^\circ)}$

Calculating $\tan(52^\circ) \approx 1.279$, we find:

$x \approx \frac{6}{1.279} \approx 4.69 \text{ cm}$

Therefore, the length of $x$ is approximately $4.69 \text{ cm}$.

Would you like further details or have any questions?

Here are some related questions to explore:

1. How would the solution change if the angle was different?
2. Can you find the hypotenuse of the triangle using the given information?
3. What is the area of the triangle?
4. How can you verify this result using the sine or cosine function?
5. How does the radius of the circle relate to the sides of the triangle?

Tip: For right triangles, always consider using trigonometric ratios (sine, cosine, tangent) to solve for unknown sides or angles.