This is a right triangle problem. To find the length of the third side, we apply the Pythagorean theorem, which states:

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

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

Given:

• One leg = $\sqrt{11}$
• Hypotenuse = $6$

We need to find the other leg, denoted as $x$.

Step 1: Apply the Pythagorean theorem

$x^2 + (\sqrt{11})^2 = 6^2$

Step 2: Simplify

$x^2 + 11 = 36$

Step 3: Solve for $x^2$

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

Step 4: Solve for $x$

$x = \sqrt{25} = 5$

Final Answer:

The length of the third side is 5.

Would you like further explanations or details? 😊

Related Questions:

1. How does the Pythagorean theorem apply to real-world problems?
2. What are some other methods to check the calculations in a right triangle?
3. How can we determine if a triangle is a right triangle given three sides?
4. How does simplifying radicals work in mathematical problems?
5. What are the differences between exact and approximate solutions in geometry?

Tip:

Always verify whether the given side lengths form a valid triangle before applying formulas.