To determine which sets of numbers could represent the lengths of the sides of a right triangle, we need to use the Pythagorean Theorem, which states:

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

Where $a$ and $b$ are the legs (the two shorter sides), and $c$ is the hypotenuse (the longest side).

Let's evaluate each set:

1. 5, 12, 13:

   • Hypotenuse = 13
   • $5^2 + 12^2 = 25 + 144 = 169$
   • $13^2 = 169$
   • Since the equation holds true, this is a valid set for a right triangle.

2. 6, 7, 8:

   • Hypotenuse = 8
   • $6^2 + 7^2 = 36 + 49 = 85$
   • $8^2 = 64$
   • Since $85 \neq 64$, this is not a valid set.

3. 16, 32, 36:

   • Hypotenuse = 36
   • $16^2 + 32^2 = 256 + 1024 = 1280$
   • $36^2 = 1296$
   • Since $1280 \neq 1296$, this is not a valid set.

4. 8, 12, 16:

   • Hypotenuse = 16
   • $8^2 + 12^2 = 64 + 144 = 208$
   • $16^2 = 256$
   • Since $208 \neq 256$, this is not a valid set.

Conclusion: The only valid set for a right triangle is 5, 12, 13.

Let me know if you'd like further details on any of these steps!

Related Questions:

1. How can we use the Pythagorean theorem to find the missing side of a right triangle?
2. What are other ways to identify right triangles besides using the Pythagorean theorem?
3. How do we determine the hypotenuse of a right triangle given the legs?
4. Can non-integer values be the sides of a right triangle? If so, give an example.
5. What is the significance of Pythagorean triples in geometry?

Tip:

Remember that for a right triangle, always ensure that the largest number is squared to check if it matches the sum of the squares of the other two sides.