To determine which set of numbers could represent the sides of a right triangle, we will apply the Pythagorean theorem: $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs and $c$ is the hypotenuse (the longest side).

Let's evaluate each option:

1. {32, 56, 65}: $32^2 + 56^2 = 1024 + 3136 = 4160$ $65^2 = 4225$ $4160 \neq 4225$, so this set does not satisfy the Pythagorean theorem.

2. {42, 56, 70}: $42^2 + 56^2 = 1764 + 3136 = 4900$ $70^2 = 4900$ $4900 = 4900$, so this set satisfies the Pythagorean theorem.

3. {44, 60, 75}: $44^2 + 60^2 = 1936 + 3600 = 5536$ $75^2 = 5625$ $5536 \neq 5625$, so this set does not satisfy the Pythagorean theorem.

4. {10, 40, 41}: $10^2 + 40^2 = 100 + 1600 = 1700$ $41^2 = 1681$ $1700 \neq 1681$, so this set does not satisfy the Pythagorean theorem.

The correct answer is {42, 56, 70}.

Would you like further details, or have any other questions?

Follow-up Questions:

1. What is the Pythagorean theorem, and why is it important?
2. How can you determine if a set of numbers represents a right triangle?
3. Can the Pythagorean theorem be used for non-right triangles?
4. What are some real-life applications of the Pythagorean theorem?
5. How do you calculate the length of a missing side of a right triangle using the Pythagorean theorem?

Tip:

Always remember that the hypotenuse is the longest side of a right triangle and opposite the right angle.