1. **Problem:** Given square FGHI with perimeter 108 units, find the length of diagonal GI. 2. **Formula:** For a square with side length $s$, perimeter $P = 4s$. 3. **Step 1:** Find side length $s$ from perimeter: $$4s = 108$$ $$s = \frac{108}{4}$$ $$s = 27$$ 4. **Step 2:** Use Pythagoras theorem to find diagonal GI in square: $$GI = \sqrt{s^2 + s^2} = \sqrt{2s^2} = s\sqrt{2}$$ 5. **Step 3:** Substitute $s=27$: $$GI = 27\sqrt{2}$$ 6. **Answer:** The length of diagonal GI is $27\sqrt{2}$ units.