1. **Problem Statement:** Determine which of the given triangles are right triangles based on their side lengths. 2. **Formula Used:** To check if a triangle is right-angled, use the Pythagorean theorem: $$a^2 + b^2 = c^2$$ where $c$ is the longest side (hypotenuse), and $a$, $b$ are the other two sides. 3. **Step-by-step Checking:** - **Triangle ABC:** Sides are 6, 9, 12. Longest side $c=12$. Calculate: $$6^2 + 9^2 = 36 + 81 = 117$$ $$12^2 = 144$$ Since $117 \neq 144$, Triangle ABC is **not** a right triangle. - **Triangle DEF:** Sides are 8, 10, 13. Longest side $c=13$. Calculate: $$8^2 + 10^2 = 64 + 100 = 164$$ $$13^2 = 169$$ Since $164 \neq 169$, Triangle DEF is **not** a right triangle. - **Triangle GHI:** Sides are 9, 12, 15. Longest side $c=15$. Calculate: $$9^2 + 12^2 = 81 + 144 = 225$$ $$15^2 = 225$$ Since $225 = 225$, Triangle GHI **is** a right triangle. - **Triangle JKL:** Sides are 10, 13, 17. Longest side $c=17$. Calculate: $$10^2 + 13^2 = 100 + 169 = 269$$ $$17^2 = 289$$ Since $269 \neq 289$, Triangle JKL is **not** a right triangle. 4. **Final Answer:** Only Triangle GHI is a right triangle.