Answer:
[tex]\sqrt{54}\; \sf cm[/tex]
Step-by-step explanation:
If we let S be the midpoint of DC, then triangle TSV is a right triangle, with the right angle at vertex S, and TV serving as its hypotenuse.
The length of TS is equal to the side length of the cube, so TS = 6 cm.
Segment SV is the hypotenuse of 45-45-90 right triangle SCV. Given that legs SC = CV = 3 cm, then according to the side length ratio of a 45-45-90 triangle, the hypotenuse SV = 3√2 cm.
Now we have the lengths of the legs of right triangle TSV, we can find the length of its hypotenuse (TV) by using the Pythagorean Theorem.
[tex]\boxed{\begin{array}{l}\underline{\textsf{Pythagorean Theorem}}\\\\c^2=a^2+b^2\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a$ and $b$ are the legs of the right triangle.}\\\phantom{ww}\bullet\;\textsf{$c$ is the hypotenuse (longest side) of the right triangle.}\\\end{array}}[/tex]
In this case:
- a = TS = 6 cm
- b = SV = 3√2 cm
- c = TV
Substitute the values into the formula and solve for TV:
[tex]TV^2=TS^2+SV^2\\\\TV^2=6^2+(3\sqrt{2})^2\\\\TV^2=36+18\\\\TV^2=54\\\\TV=\sqrt{54}\; \sf cm[/tex]
Therefore, the distance from T to V is:
[tex]\Large\boxed{\boxed{\sqrt{54}\; \sf cm}}[/tex]
[tex]\dotfill[/tex]
Additional Notes
The other way to solve this is to use the diagonal of cuboid formula, which is the square root of the sum of the squares of the cuboid's length, breadth and height.
[tex]d=\sqrt{l^2+b^2+h^2}[/tex]
In this case, l = 3 cm, b = 3 cm, and h = 6 cm:
[tex]TV=\sqrt{3^2+3^2+6^2}\\\\TV=\sqrt{9+9+36}\\\\TV=\sqrt{54}\; \sf cm[/tex]
