To find the length of the line segment with endpoints (6, 3) and (6, 7), we first calculate its original length.
The length of a line segment can be determined using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the endpoints (6, 3) and (6, 7):
- \(x_1 = 6, y_1 = 3\)
- \(x_2 = 6, y_2 = 7\)
Now, we calculate the length: \[ d = \sqrt{(6 - 6)^2 + (7 - 3)^2} = \sqrt{0 + (4)^2} = \sqrt{16} = 4 \text{ units} \]
Next, we consider the translation. Translating the line segment 3 units up and 4 units to the right means we add these values to the coordinates of both endpoints.
The new endpoints after translation are:
- From (6, 3) to \((6 + 4, 3 + 3) = (10, 6)\)
- From (6, 7) to \((6 + 4, 7 + 3) = (10, 10)\)
Now, we calculate the length of the translated segment: Using the new endpoints (10, 6) and (10, 10):
\[ d = \sqrt{(10 - 10)^2 + (10 - 6)^2} = \sqrt{0 + (4)^2} = \sqrt{16} = 4 \text{ units} \]
Thus, the length of the line segment after the translation remains 4 units.
The correct response is: 4 units.