To determine the length of the line segment after the translation, we first find the length of the original line segment between the endpoints (6, 3) and (6, 7).
The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For the given endpoints (6, 3) and (6, 7):
- \( x_1 = 6 \), \( y_1 = 3 \)
- \( x_2 = 6 \), \( y_2 = 7 \)
Plugging these values into the distance formula:
\[ d = \sqrt{(6 - 6)^2 + (7 - 3)^2} = \sqrt{0 + 16} = \sqrt{16} = 4 \]
The length of the line segment before translation is 4 units.
Next, we will perform the translation of 3 units up and 4 units to the right:
-
Translating (6, 3):
- New x-coordinate: \( 6 + 4 = 10 \)
- New y-coordinate: \( 3 + 3 = 6 \)
- New point: (10, 6)
-
Translating (6, 7):
- New x-coordinate: \( 6 + 4 = 10 \)
- New y-coordinate: \( 7 + 3 = 10 \)
- New point: (10, 10)
Now we find the length of the new line segment connecting the points (10, 6) and (10, 10):
Using the distance formula again:
\[ d' = \sqrt{(10 - 10)^2 + (10 - 6)^2} = \sqrt{0 + 16} = \sqrt{16} = 4 \]
The length of the line segment remains 4 units after the translation.
Thus, the final answer is:
4 units