1. **State the problem:** We have a rectangular prism ABCDEFGH with the sum of all edge lengths equal to 28 cm and total surface area equal to 13 cm². We need to find the length of the diagonal EC. 2. **Define variables:** Let the edges of the prism be $a$, $b$, and $c$. 3. **Sum of edges:** A rectangular prism has 12 edges: 4 edges of length $a$, 4 of length $b$, and 4 of length $c$. So, $$4(a+b+c) = 28$$ Divide both sides by 4: $$\cancel{4}(a+b+c) = \cancel{4}7 \implies a+b+c=7$$ 4. **Surface area:** The surface area $S$ of a rectangular prism is $$S = 2(ab + bc + ac) = 13$$ Divide both sides by 2: $$ab + bc + ac = \frac{13}{2} = 6.5$$ 5. **Diagonal length:** The diagonal $EC$ connects opposite corners of the prism. Its length is given by $$EC = \sqrt{a^2 + b^2 + c^2}$$ 6. **Use identity:** Recall that $$(a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ac)$$ Substitute known values: $$7^2 = a^2 + b^2 + c^2 + 2(6.5)$$ $$49 = a^2 + b^2 + c^2 + 13$$ 7. **Solve for $a^2 + b^2 + c^2$:** $$a^2 + b^2 + c^2 = 49 - 13 = 36$$ 8. **Find diagonal length:** $$EC = \sqrt{36} = 6$$ **Final answer:** The length of diagonal $EC$ is 6 cm.