To determine which set of side lengths can be used to form a right triangle, we apply the Pythagorean theorem. The theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides:

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

where $a$, $b$, and $c$ are the side lengths, with $c$ being the hypotenuse (the largest value).

Let's check each option:

1. 30, 40, 50: $30^2 + 40^2 = 50^2$ $900 + 1600 = 2500$ $2500 = 2500$ This is a right triangle.

2. 4, 5, 6: $4^2 + 5^2 = 6^2$ $16 + 25 = 36$ $41 \neq 36$ This is not a right triangle.

3. 2, 3, 4: $2^2 + 3^2 = 4^2$ $4 + 9 = 16$ $13 \neq 16$ This is not a right triangle.

4. 10, 20, 30: $10^2 + 20^2 = 30^2$ $100 + 400 = 900$ $500 \neq 900$ This is not a right triangle.

Conclusion:

The set that forms a right triangle is 30, 40, 50.

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. How can we derive the Pythagorean triples?
2. What are the properties of similar triangles in terms of side lengths?
3. Can side lengths of a non-right triangle still satisfy other mathematical properties?
4. How can we generalize the Pythagorean theorem to higher dimensions?
5. Are there real-world applications of the Pythagorean theorem?

Tip: Always verify the largest side as the hypotenuse when using the Pythagorean theorem.