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To find the length of the line segment after the translation, we need to find the distance between the translated endpoints.
After being translated 3 units up and 4 units to the right, the new endpoints will be:
Endpoint 1: (6 + 4, 3 + 3) = (10, 6)
Endpoint 2: (6 + 4, 7 + 3) = (10, 10)
Using the distance formula, we can find the length of the line segment:
Length = √[(10 - 10)^2 + (10 - 6)^2]
= √[0 + 16]
= √16
= 4 units
Therefore, the length of the line segment after the translation is 4 units.