1. **Problem:** Determine which combinations of sides are Pythagorean triples. 2. **Definition:** A Pythagorean triple consists of three positive integers $a$, $b$, and $c$ such that $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse (the longest side). 3. **Check each option:** - a. $6, 12, 13$ $$6^2 + 12^2 = 36 + 144 = 180$$ $$13^2 = 169$$ Since $180 \neq 169$, this is not a Pythagorean triple. - b. $3, 4, 5$ $$3^2 + 4^2 = 9 + 16 = 25$$ $$5^2 = 25$$ Since $25 = 25$, this is a Pythagorean triple. - c. $20, 24, 25$ $$20^2 + 24^2 = 400 + 576 = 976$$ $$25^2 = 625$$ Since $976 \neq 625$, this is not a Pythagorean triple. - d. $8, 15, 17$ $$8^2 + 15^2 = 64 + 225 = 289$$ $$17^2 = 289$$ Since $289 = 289$, this is a Pythagorean triple. **Final answer:** The Pythagorean triples are options b and d.