Question

if a line segment with endpoints (6, 3) and (6, 7) is translated 3 units up 4 units to the right, how long is the line segment after the translation?

a. there is not enough information to measure the line segment
b. 0 units
c. 10 units
d. 4 units

Answer 1

To find the length of the line segment after the translation, we first need to calculate the original length of the line segment with endpoints (6, 3) and (6, 7).

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

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

For the 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, we have:

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

Next, we translate the endpoints of this line segment. The translation moves the segment 3 units up and 4 units to the right.

New coordinates after translation:

1. The point (6, 3) becomes:
- \(x = 6 + 4 = 10\)
- \(y = 3 + 3 = 6\)
- So, the new point is (10, 6).

2. The point (6, 7) becomes:
- \(x = 6 + 4 = 10\)
- \(y = 7 + 3 = 10\)
- So, the new point is (10, 10).

Now, we can calculate the new length of the line segment with its endpoints at (10, 6) and (10, 10):

Using the distance formula again:

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

For the new endpoints (10, 6) and (10, 10):

- \(x_1 = 10\)
- \(y_1 = 6\)
- \(x_2 = 10\)
- \(y_2 = 10\)

Calculating the distance:

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

Thus, the length of the line segment after the translation is **4 units**.

The correct answer is **d. 4 units**.