To find the length of the line segment after translation, we first need to compute its initial length.
The endpoints of the line segment are given as \((6, 3)\) and \((6, 7)\). We can determine the length of the line segment using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of our endpoints \((x_1, y_1) = (6, 3)\) and \((x_2, y_2) = (6, 7)\):
\[ d = \sqrt{(6 - 6)^2 + (7 - 3)^2} = \sqrt{0^2 + 4^2} = \sqrt{0 + 16} = \sqrt{16} = 4 \]
So the length of the line segment before translation is 4 units.
Next, we translate the line segment 3 units up and 4 units to the right. This translation changes the coordinates of each endpoint as follows:
-
Translating \((6, 3)\):
- Move 4 units right: \(6 + 4 = 10\)
- Move 3 units up: \(3 + 3 = 6\)
- New coordinates: \((10, 6)\)
-
Translating \((6, 7)\):
- Move 4 units right: \(6 + 4 = 10\)
- Move 3 units up: \(7 + 3 = 10\)
- New coordinates: \((10, 10)\)
Now, the new endpoints of the translated line segment are \((10, 6)\) and \((10, 10)\). Let's find the length of this new segment using the same distance formula:
\[ d = \sqrt{(10 - 10)^2 + (10 - 6)^2} = \sqrt{0^2 + 4^2} = \sqrt{0 + 16} = \sqrt{16} = 4 \]
Thus, the length of the line segment after the translation is still \(4\) units.
Therefore, the final answer is:
\[ \boxed{4} \]