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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?

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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.

Which set of side lengths can be used to form a right triangle?

Learn which set of side lengths can form a right triangle using the Pythagorean theorem. This problem explores the relationship between the sides of a right triangle, including sets like 30, 40, 50. Suitable for students in Grades 6-8.

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