Answer:
MN = sqrt(34)
Explanation:
First, draw the segment FN. The diagonal, MN, of the prism is the hypotenuse of triangle NFM. Triangle NFM is a right triangle with legs FN and FM and hypotenuse MN.
Leg FM of triangle NFM has length 4 cm.
We need to find the length of leg FN.
Look at the base of the prism which is square UNAF. FN is a diagonal of that square. Now think of right triangle FUN with legs UN and UF, each of length 3 cm. We can find FN with the Pythagorean theorem.
(UF)^2 + (UN)^2 = (FN)^2
3^2 + 3^2 = (FN)^2
(FN)^2 = 18
FN = sqrt(18)
Now we know FN. We use FN and FM as legs and find MN, the hypotenuse of triangle NFM.
(FN)^2 + (FM)^2 = (MN)^2
18 + 4^2 = (MN)^2
18 + 16 = (MN)^2
(MN)^2 = 34
MN = sqrt(34)