The problem involves a right triangle, and we need to find the value of $x$. The sides of the triangle are labeled as follows:

• The hypotenuse (the side opposite the right angle) is $x$.
• One leg of the triangle is $8$.
• The other leg of the triangle is $x - 4$.

We can use the Pythagorean theorem to solve for $x$. The Pythagorean theorem states:

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

Substituting the given values into the equation:

$x^2 = 8^2 + (x - 4)^2$

Now, let's solve for $x$:

1. Expand the equation: $x^2 = 64 + (x^2 - 8x + 16)$

2. Simplify the equation: $x^2 = x^2 - 8x + 80$

3. Subtract $x^2$ from both sides: $0 = -8x + 80$

4. Solve for $x$: $8x = 80$ $x = \frac{80}{8} = 10$

So, the value of $x$ is 10.

Would you like more details on this solution or have any other questions?

Related Questions:

1. How does the Pythagorean theorem work in non-right triangles?
2. What happens if both legs of the triangle were given and we needed to find the hypotenuse?
3. Can the value of $x$ ever be negative in such problems?
4. How can we verify the solution in the context of the triangle's dimensions?
5. What if the triangle's dimensions were based on real-life measurements? How would rounding affect the answer?

Tip: Always check the context of the problem to ensure the triangle is right-angled before applying the Pythagorean theorem.