This image shows a right triangle with the following side lengths:

• The hypotenuse is labeled $x + 2$,
• One leg is labeled $x$,
• The other leg is labeled $\sqrt{10}$.

We can apply the Pythagorean theorem to find the value of $x$. The Pythagorean theorem states that, for a right triangle:

$\text{(Leg 1)}^2 + \text{(Leg 2)}^2 = \text{Hypotenuse}^2$

Substituting the given side lengths into this equation:

$x^2 + (\sqrt{10})^2 = (x + 2)^2$

This simplifies to:

$x^2 + 10 = (x + 2)^2$

Expanding the right-hand side:

$x^2 + 10 = x^2 + 4x + 4$

Now, subtract $x^2$ from both sides:

$10 = 4x + 4$

Subtract 4 from both sides:

$6 = 4x$

Divide by 4:

$x = \frac{6}{4} = 1.5$

Thus, the value of $x$ is $1.5$.

Would you like a detailed explanation of any part of the solution?

Here are some follow-up questions:

1. How does the Pythagorean theorem apply to all right triangles?
2. What are some examples of common right triangles?
3. Can we apply this method to an obtuse triangle? Why or why not?
4. How do the properties of a square root affect solving the equation?
5. What other methods exist for solving a triangle with unknown sides?

Tip: Always verify your result by substituting the value of $x$ back into the original triangle dimensions to check for consistency.