Question

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?
1. 4 units
2. 0 units
3. 10 units
4. There is not enough information to measure the line segment.

Answer 1

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 \((6, 7)\).

The length of a line segment can be calculated using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

For the original endpoints:
- \((x_1, y_1) = (6, 3)\)
- \((x_2, y_2) = (6, 7)\)

Now we substitute in the values:

\[
d = \sqrt{(6 - 6)^2 + (7 - 3)^2} = \sqrt{0^2 + 4^2} = \sqrt{16} = 4
\]

So the length of the original line segment is 4 units.

Now we translate the endpoints of the line segment 3 units up and 4 units to the right.

- The point \((6, 3)\) after translation becomes:
\((6 + 4, 3 + 3) = (10, 6)\)

- The point \((6, 7)\) after translation becomes:
\((6 + 4, 7 + 3) = (10, 10)\)

Now we have the new endpoints \((10, 6)\) and \((10, 10)\).

We can calculate the length of the translated line segment in the same way:

\[
d = \sqrt{(10 - 10)^2 + (10 - 6)^2} = \sqrt{0^2 + 4^2} = \sqrt{16} = 4
\]

So the length of the line segment after translation remains **4** units.

Thus, the answer is:

1. 4 units