Given that the segment MK is 7 units in length and the segment MN is 6 units in length, the ratio of the lengths MK to MN is 7:4.

To find the length of the segment PN, we can set up a proportion using the given ratio:

MK/MN = 7/4

Let x be the length of PN:

MK/PN = 7/x

Since MN = MK - PN, we can substitute the values we know:

7/4 = 7/(6+x)

Now we can solve for x:

7(6+x) = 4(7)
42 + 7x = 28
7x = -14
x = -2

Therefore, the length of segment PN is -2 units. However, since length cannot be negative, it seems there might be an error in the calculations. Please check the data and calculations again to verify the answer.