To find the length $x$ in the right triangle, we can use the Pythagorean Theorem, which states:

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

where:

• $a$ and $b$ are the lengths of the legs of the triangle, and
• $c$ is the length of the hypotenuse (the side opposite the right angle).

In this triangle:

• The hypotenuse is $6$,
• One leg is $5$, and
• The other leg (which we are trying to find) is $x$.

Applying the Pythagorean Theorem:

$5^2 + x^2 = 6^2$

Calculating each term:

$25 + x^2 = 36$

Now, isolate $x^2$ by subtracting $25$ from both sides:

$x^2 = 36 - 25$ $x^2 = 11$

Taking the square root of both sides:

$x = \sqrt{11}$

So, the length $x$ is approximately:

$x \approx 3.32$

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

Here are five related questions that may interest you:

1. How would we solve this problem if we were given the length of $x$ and needed to find the hypotenuse?
2. How is the Pythagorean Theorem applied when given only one leg and the hypotenuse?
3. What are the steps for solving right triangles with trigonometric ratios instead?
4. How can we confirm our answer by using trigonometry?
5. How is this theorem used in three-dimensional geometry?

Tip: When dealing with right triangles, remember that the hypotenuse is always the longest side, opposite the right angle.