To solve for $x$ in the given right triangle, we'll use the Pythagorean theorem. The image shows a right triangle with one leg labeled $12$ and the hypotenuse labeled $13$.

The Pythagorean theorem states: $a^2 + b^2 = c^2$

Here, $a$ is the unknown leg (let's call it $x$), $b = 12$, and $c = 13$.

Substitute the given values into the equation: $x^2 + 12^2 = 13^2$

Calculate the squares: $x^2 + 144 = 169$

Subtract 144 from both sides to solve for $x^2$: $x^2 = 169 - 144$ $x^2 = 25$

Take the square root of both sides: $x = \sqrt{25}$ $x = 5$

So, $x = 5$.

Would you like more details or have any questions?

Here are some related questions:

1. How can we verify the solution for $x$?
2. What is the Pythagorean theorem and how is it used in different types of triangles?
3. Can you explain why the Pythagorean theorem only applies to right triangles?
4. What are the properties of a 3-4-5 triangle?
5. How do you find the length of a hypotenuse if both legs of a right triangle are known?
6. What is the significance of the Pythagorean theorem in real-life applications?
7. How would you solve for $x$ if the hypotenuse and one leg were given in terms of variables?
8. How can trigonometric ratios be used to solve right triangle problems?

Tip: When working with right triangles, always check if the given lengths form a Pythagorean triple (like 3-4-5 or 5-12-13) to simplify calculations.