We are given a right triangle with one side measuring $\sqrt{21}$ and the hypotenuse measuring 5. We need to find the length of the third side in simplest radical form.

We can use the Pythagorean theorem to solve this. The theorem states that in a right triangle:

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

Where $a$ and $b$ are the legs, and $c$ is the hypotenuse.

In this case, one leg is $\sqrt{21}$, and the hypotenuse is 5. Let's denote the unknown side as $x$. Using the Pythagorean theorem:

$(\sqrt{21})^2 + x^2 = 5^2$

Simplifying the equation:

$21 + x^2 = 25$

Now, solve for $x^2$:

$x^2 = 25 - 21$ $x^2 = 4$

Taking the square root of both sides:

$x = \sqrt{4} = 2$

Thus, the length of the third side is 2.

Let me know if you'd like further clarification or have more questions!

Here are 5 related questions you can explore:

1. How is the Pythagorean theorem derived from Euclidean geometry?
2. What are other methods for solving for a missing side in right triangles?
3. How do you simplify square roots like $\sqrt{21}$?
4. Can the Pythagorean theorem be extended to non-Euclidean geometry?
5. How do trigonometric ratios relate to side lengths in a right triangle?

Tip: Always double-check that you are applying the Pythagorean theorem to the correct sides—especially verifying the hypotenuse!