If a line segment with endpoints (6,3) and (6,7) is translated 3 units up and 4 units to the right, how long is the line segment after the translation?
To determine the length of the line segment after translation, we can use the distance formula.
The formula for the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:
$$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
In this case, the initial endpoints are (6,3) and (6,7). After the translation, the new endpoints will be (6 + 4, 3 + 3) = (10, 6) and (6 + 4, 7 + 3) = (10, 10).
Using the distance formula, we can find the length of the line segment:
$$\text{Distance} = \sqrt{(10 - 10)^2 + (10 - 6)^2}$$
$$ = \sqrt{0 + 16}$$
$$ = \sqrt{16}$$
$$ = 4$$
Therefore, the length of the line segment after the translation is 4 units.