To find the length of the line segment after translation, we first determine the original coordinates of the endpoints of the line segment and then apply the translation.
The endpoints of the line segment are:
- \( A(6, 3) \)
- \( B(6, 7) \)
Now, we can calculate the length of the original line segment using the distance formula:
\[ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates of points A and B into the formula:
\[ \text{Length} = \sqrt{(6 - 6)^2 + (7 - 3)^2} = \sqrt{(0)^2 + (4)^2} = \sqrt{16} = 4 \]
Now, we apply the translation:
- We move the segment 3 units up, which will increase the y-coordinates by 3.
- We move the segment 4 units to the right, which will increase the x-coordinates by 4.
Thus the new coordinates of the endpoints will be:
- For point A: \( (6 + 4, 3 + 3) = (10, 6) \)
- For point B: \( (6 + 4, 7 + 3) = (10, 10) \)
Now we compute the length of the line segment after translation using the new coordinates:
\[ \text{New Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the new coordinates:
\[ \text{New Length} = \sqrt{(10 - 10)^2 + (10 - 6)^2} = \sqrt{(0)^2 + (4)^2} = \sqrt{16} = 4 \]
Therefore, the length of the line segment after the translation is still \( \boxed{4} \).
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