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To determine which of the given sets of numbers could represent the three sides of a right triangle, we can apply the **Pythagorean Theorem**. The theorem states that for any right triangle with sides \( a \), \( b \), and \( c \) (where \( c \) is the hypotenuse), the relationship \( a^2 + b^2 = c^2 \) must hold.

Let's check each set to see if it satisfies this condition:

1. **Set {20, 21, 28}**:
   - Check if \( 20^2 + 21^2 = 28^2 \):
     \[
     20^2 + 21^2 = 400 + 441 = 841
     \]
     \[
     28^2 = 784
     \]
     This does **not** satisfy the equation, so this is not a right triangle.

2. **Set {24, 32, 39}**:
   - Check if \( 24^2 + 32^2 = 39^2 \):
     \[
     24^2 + 32^2 = 576 + 1024 = 1600
     \]
     \[
     39^2 = 1521
     \]
     This does **not** satisfy the equation, so this is not a right triangle.

3. **Set {31, 60, 68}**:
   - Check if \( 31^2 + 60^2 = 68^2 \):
     \[
     31^2 + 60^2 = 961 + 3600 = 4561
     \]
     \[
     68^2 = 4624
     \]
     This does **not** satisfy the equation, so this is not a right triangle.

4. **Set {6, 8, 10}**:
   - Check if \( 6^2 + 8^2 = 10^2 \):
     \[
     6^2 + 8^2 = 36 + 64 = 100
     \]
     \[
     10^2 = 100
     \]
     This satisfies the Pythagorean Theorem, so this set **does** represent the sides of a right triangle.

Thus, the correct answer is **{6, 8, 10}**.

Would you like further explanation or details on the process? Here are five related questions to deepen your understanding:

1. How can the Pythagorean Theorem be used to determine whether any set of numbers forms a right triangle?
2. What are Pythagorean triples, and why are they important in geometry?
3. Can a triangle with irrational numbers as side lengths be a right triangle?
4. What other properties of right triangles can be derived using the Pythagorean Theorem?
5. How do you determine which number should be the hypotenuse when using the Pythagorean Theorem?

**Tip:** In a right triangle, the hypotenuse is always the longest side.

Which of the following sets of numbers could represent the three sides of a right triangle?

Determine which set of numbers could represent the sides of a right triangle by applying the Pythagorean Theorem. This problem explores fundamental concepts in geometry and trigonometry, suitable for students in Grades 8-10.

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