1. **Problem statement:** We have a cube ABCDEFGH with side length 6 cm. Points T and V are midpoints of edges AB and CH respectively. 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. **Coordinates assignment:** Assign coordinates to vertices for easier calculation. Let A = (0,0,0), B = (6,0,0), C = (6,6,0), H = (6,6,6). 3. **Find coordinates of T and V:** T is midpoint of AB: $T = \left(\frac{0+6}{2}, \frac{0+0}{2}, \frac{0+0}{2}\right) = (3,0,0)$. V is midpoint of CH: $V = \left(\frac{6+6}{2}, \frac{6+6}{2}, \frac{0+6}{2}\right) = (6,6,3)$. 4. **Distance formula:** Distance $d$ between points $T(x_1,y_1,z_1)$ and $V(x_2,y_2,z_2)$ is $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$ 5. **Calculate distance:** $$d = \sqrt{(6-3)^2 + (6-0)^2 + (3-0)^2} = \sqrt{3^2 + 6^2 + 3^2} = \sqrt{9 + 36 + 9} = \sqrt{54}$$ 6. **Simplify $\sqrt{54}$:** $$\sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6}$$ **Final answer:** The distance from T to V is $3\sqrt{6}$ cm. This method uses coordinate geometry for a straightforward calculation inside the cube.