To determine which set of numbers represents the sides of a right triangle, we need to check if they satisfy the Pythagorean theorem:

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

where $a$, $b$, and $c$ are the lengths of the triangle's sides, and $c$ is the hypotenuse (the largest number in the set).

Calculations for each set:

1. Set $\{36, 84, 91\}$: $36^2 + 84^2 = 1296 + 7056 = 8352, \quad 91^2 = 8281$ $8352 \neq 8281$, so this is not a right triangle.

2. Set $\{8, 24, 25\}$: $8^2 + 24^2 = 64 + 576 = 640, \quad 25^2 = 625$ $640 \neq 625$, so this is not a right triangle.

3. Set $\{32, 60, 68\}$: $32^2 + 60^2 = 1024 + 3600 = 4624, \quad 68^2 = 4624$ $4624 = 4624$, so this is a right triangle.

4. Set $\{51, 67, 85\}$: $51^2 + 67^2 = 2601 + 4489 = 7090, \quad 85^2 = 7225$ $7090 \neq 7225$, so this is not a right triangle.

Answer:

The set $\{32, 60, 68\}$ represents the sides of a right triangle.

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Would you like further details or have any questions? Here are some follow-up topics:

1. How to identify right triangles using Pythagorean triples?
2. What are primitive Pythagorean triples?
3. How to find the hypotenuse of a triangle given two sides?
4. How does the Pythagorean theorem apply in three dimensions?
5. Can a triangle with irrational sides be a right triangle?

Tip: Always check that the largest side in the set is the hypotenuse when applying the Pythagorean theorem.