To find the length $x$ in the given right triangle, we can use the Pythagorean theorem:

$a^2 + b^2 = c^2$

In this case:

• One leg is $x$,
• The other leg is $4$,
• The hypotenuse is $8$.

Substitute these values:

$x^2 + 4^2 = 8^2$

Calculate each term:

1. $4^2 = 16$
2. $8^2 = 64$

So we have:

$x^2 + 16 = 64$

Now, isolate $x^2$:

$x^2 = 64 - 16$ $x^2 = 48$

Take the square root of both sides:

$x = \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$

Thus, the length $x$ is $4\sqrt{3}$.

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

Here are some related questions for further exploration:

1. What if the hypotenuse was a different length, say 10, how would $x$ change?
2. How do we find the exact decimal value of $4\sqrt{3}$?
3. Can the Pythagorean theorem be applied to non-right triangles?
4. How can trigonometry be used to verify the side lengths of a right triangle?
5. How would the calculation change if we were solving for the other leg instead?

Tip: For right triangles, always ensure that the hypotenuse is the longest side when applying the Pythagorean theorem.