1. **State the problem:** Determine which sets of lengths $(a,b,c)$ can form a right triangle. 2. **Formula used:** For a triangle with sides $a$, $b$, and $c$ (where $c$ is the longest side), the Pythagorean theorem states: $$a^2 + b^2 = c^2$$ If this holds true, the triangle is right-angled. 3. **Check each set:** - For $(5,12,13)$: $$5^2 + 12^2 = 25 + 144 = 169$$ $$13^2 = 169$$ Since $169 = 169$, yes, it forms a right triangle. - For $(8,10,12)$: $$8^2 + 10^2 = 64 + 100 = 164$$ $$12^2 = 144$$ Since $164 \neq 144$, no right triangle. - For $(9,12,15)$: $$9^2 + 12^2 = 81 + 144 = 225$$ $$15^2 = 225$$ Since $225 = 225$, yes, it forms a right triangle. - For $(12,16,20)$: $$12^2 + 16^2 = 144 + 256 = 400$$ $$20^2 = 400$$ Since $400 = 400$, yes, it forms a right triangle. - For $(15,16,17)$: $$15^2 + 16^2 = 225 + 256 = 481$$ $$17^2 = 289$$ Since $481 \neq 289$, no right triangle. - For $(20,40,60)$: $$20^2 + 40^2 = 400 + 1600 = 2000$$ $$60^2 = 3600$$ Since $2000 \neq 3600$, no right triangle. **Final answers:** - $(5,12,13)$: Yes - $(8,10,12)$: No - $(9,12,15)$: Yes - $(12,16,20)$: Yes - $(15,16,17)$: No - $(20,40,60)$: No