To find the length of the line segment, we first need to determine its original length. The endpoints of the line segment are (6, 3) and (6, 7).
The length of a vertical line segment can be calculated using the formula for the distance between two points, specifically focusing on their y-coordinates since the x-coordinates are the same:
\[ \text{Length} = |y_2 - y_1| = |7 - 3| = 4 \text{ units} \]
Next, we will apply the translation that moves the line segment 3 units up and 4 units to the right. This will change the endpoints:
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The original point (6, 3) becomes: \[ (6 + 4, 3 + 3) = (10, 6) \]
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The original point (6, 7) becomes: \[ (6 + 4, 7 + 3) = (10, 10) \]
Now the new endpoints after the translation are (10, 6) and (10, 10). We can again calculate the length of this new line segment:
\[ \text{New Length} = |y_2 - y_1| = |10 - 6| = 4 \text{ units} \]
So, after the translation, the length of the line segment remains 4 units.
Therefore, the answer is:
4 units