1. **Problem statement:** We have a cube ABCDEFGH with side length 6 cm. T is the midpoint of edge AB, and V is the midpoint of edge CH. We need to find the straight-line distance from T to V inside the cube, expressed as $\sqrt{a}$ cm where $a$ is an integer. 2. **Set up coordinates:** Assign coordinates to the cube vertices for easier calculation. Let: - $A = (0,0,6)$ - $B = (6,0,6)$ - $C = (6,6,0)$ - $D = (0,6,0)$ - $E = (0,0,0)$ - $F = (0,6,6)$ - $G = (6,6,6)$ - $H = (6,0,0)$ 3. **Find coordinates of T and V:** - T is midpoint of AB: $$T = \left(\frac{0+6}{2}, \frac{0+0}{2}, \frac{6+6}{2}\right) = (3,0,6)$$ - V is midpoint of CH: $$V = \left(\frac{6+6}{2}, \frac{6+0}{2}, \frac{0+0}{2}\right) = (6,3,0)$$ 4. **Calculate distance TV:** Use distance formula: $$TV = \sqrt{(6-3)^2 + (3-0)^2 + (0-6)^2}$$ $$= \sqrt{3^2 + 3^2 + (-6)^2}$$ $$= \sqrt{9 + 9 + 36}$$ $$= \sqrt{54}$$ 5. **Simplify the radical:** $$\sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6}$$ **Final answer:** The distance from T to V is $\boxed{3\sqrt{6}}$ cm.