Question

Consider a line segment with endpoints (4, 7) and (1, 11).

Which line segment is equal in distance to the given line segment?
Responses
A A line segment with endpoints (−6, 4) and (2, −5).A line segment with endpoints (−6, 4) and (2, −5).
B A line segment with endpoints (1, −4) and (9, 2).A line segment with endpoints (1, −4) and (9, 2).
C A line segment with endpoints (−3, 1) and (4, 5).A line segment with endpoints (−3, 1) and (4, 5).
D A line segment with endpoints (5, 3) and (1, 6).

Answer 1

To determine which line segment is equal in distance to the given line segment with endpoints (4, 7) and (1, 11), we will first calculate the distance of the initial line segment using the distance formula:

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

For the line segment with endpoints (4, 7) and (1, 11):

- \(x_1 = 4\), \(y_1 = 7\)
- \(x_2 = 1\), \(y_2 = 11\)

Substituting these values into the distance formula:

\[
d = \sqrt{(1 - 4)^2 + (11 - 7)^2}
\]
\[
= \sqrt{(-3)^2 + (4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= \sqrt{25}
\]
\[
= 5
\]

Now we will calculate the distance for each response option to find a matching length of 5.

**Option A:** Endpoints (−6, 4) and (2, −5)

\[
d = \sqrt{(2 - (-6))^2 + (-5 - 4)^2}
\]
\[
= \sqrt{(2 + 6)^2 + (-9)^2}
\]
\[
= \sqrt{8^2 + 9^2}
\]
\[
= \sqrt{64 + 81}
\]
\[
= \sqrt{145} \text{ (not equal to 5)}
\]

**Option B:** Endpoints (1, −4) and (9, 2)

\[
d = \sqrt{(9 - 1)^2 + (2 - (-4))^2}
\]
\[
= \sqrt{(8)^2 + (6)^2}
\]
\[
= \sqrt{64 + 36}
\]
\[
= \sqrt{100}
\]
\[
= 10 \text{ (not equal to 5)}
\]

**Option C:** Endpoints (−3, 1) and (4, 5)

\[
d = \sqrt{(4 - (-3))^2 + (5 - 1)^2}
\]
\[
= \sqrt{(4 + 3)^2 + (4)^2}
\]
\[
= \sqrt{7^2 + 4^2}
\]
\[
= \sqrt{49 + 16}
\]
\[
= \sqrt{65} \text{ (not equal to 5)}
\]

**Option D:** Endpoints (5, 3) and (1, 6)

\[
d = \sqrt{(1 - 5)^2 + (6 - 3)^2}
\]
\[
= \sqrt{(-4)^2 + (3)^2}
\]
\[
= \sqrt{16 + 9}
\]
\[
= \sqrt{25}
\]
\[
= 5 \text{ (equal to 5)}
\]

Therefore, the line segment that is equal in distance to the given line segment is:

**D: A line segment with endpoints (5, 3) and (1, 6).**