To determine the length of the line segment after the translation, we first need to calculate the length of the original line segment with endpoints (6, 3) and (6, 7).
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Calculating the length of the original line segment: The length of a vertical line segment can be found by subtracting the y-coordinates of the endpoints. Here, the endpoints are (6, 3) and (6, 7):
\[ \text{Length} = |y_2 - y_1| = |7 - 3| = |4| = 4 , \text{units} \]
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Applying the translation: The translation moves the line segment 3 units up and 4 units to the right. This means we need to update the coordinates of both endpoints.
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The new coordinates of the first endpoint (6, 3) after translation:
- \(x\): 6 + 4 = 10
- \(y\): 3 + 3 = 6
- New point: (10, 6)
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The new coordinates of the second endpoint (6, 7) after translation:
- \(x\): 6 + 4 = 10
- \(y\): 7 + 3 = 10
- New point: (10, 10)
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Calculating the length of the translated line segment: The new endpoints of the line segment are (10, 6) and (10, 10). The length remains the same as the original length since translation does not affect the distance between points. We calculate it again:
\[ \text{Length} = |10 - 6| = |10 - 6| = |4| = 4 , \text{units} \]
Therefore, the length of the line segment after the translation is 4 units.