1. **Problem:** Calculate the length of AG in the first figure (a prism with base AB = 8, height 8, depth 8). 2. **Formula and rules:** To find the length of a diagonal in a rectangular prism, use the 3D distance formula or Pythagoras theorem in 3 steps: $$AG = \sqrt{AB^2 + BG^2}$$ where $BG$ is the diagonal of the base rectangle. 3. **Step 1:** Calculate the diagonal of the base rectangle (square with side 8): $$BG = \sqrt{AB^2 + BC^2} = \sqrt{8^2 + 8^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2}$$ 4. **Step 2:** Now calculate $AG$ using the height (8) and base diagonal $BG$: $$AG = \sqrt{BG^2 + height^2} = \sqrt{(8\sqrt{2})^2 + 8^2} = \sqrt{128 + 64} = \sqrt{192}$$ 5. **Step 3:** Simplify $\sqrt{192}$: $$\sqrt{192} = \sqrt{64 \times 3} = 8\sqrt{3}$$ **Final answer:** $$AG = 8\sqrt{3}$$ This means the length of AG is $8\sqrt{3}$ units. This explanation covers the first question only as per instructions.