Question

Demonstrate the translation of a line segment of 3 units to the left and 5 units down if the endpoints of the line segment are (−2, 7)
and (0, 3)
.(1 point)
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 point)
A line segment has endpoints (1, 1)
and (4, 4)
. Demonstrating the properties of reflections across the y
-axis, what are the reflected endpoints?(1 point)
A line contains the points (6,−3)
and (3, 1)
. When this line is reflected across the y
-axis, what are the two new points?(1 point)
A line segment has endpoints (0, 5)
and (6, 5)
. After the line segment is reflected across the x
-axis, how long will it be?(1 point

Answer 1

Let's address each of the questions step by step.

### Question 1: Translation of a Line Segment

Given points:
\( A(-2, 7) \) and \( B(0, 3) \).

To translate the line segment 3 units to the left and 5 units down, we adjust the coordinates of each endpoint as follows:

1. **Translate Point A:**
\[
A' = (x - 3, y - 5) = (-2 - 3, 7 - 5) = (-5, 2)
\]

2. **Translate Point B:**
\[
B' = (x - 3, y - 5) = (0 - 3, 3 - 5) = (-3, -2)
\]

So, the new endpoints after translation are \( A'(-5, 2) \) and \( B'(-3, -2) \).

---

### Question 2: Length of a Line Segment After Translation

Given points:
\( P(6, 3) \) and \( Q(6, 7) \).

First, we need to find the length of the line segment \( PQ \) before translation. The formula for the length \( L \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting our points into the formula:
\[
L = \sqrt{(6 - 6)^2 + (7 - 3)^2} = \sqrt{0 + 16} = \sqrt{16} = 4
\]

Now, when the segment is translated 3 units up and 4 units to the right, the new coordinates will be:
- \( P'(6 + 4, 3 + 3) = (10, 6) \)
- \( Q'(6 + 4, 7 + 3) = (10, 10) \)

The length remains unchanged, so the length of the segment after translation is still 4.

---

### Question 3: Reflection Across the y-axis

Given endpoints: \( R(1, 1) \) and \( S(4, 4) \).

To reflect a point across the y-axis, we change the sign of the x-coordinate:
1. **Reflect Point R:**
\[
R' = (-1, 1)
\]

2. **Reflect Point S:**
\[
S' = (-4, 4)
\]

So reflected endpoints are \( R'(-1, 1) \) and \( S'(-4, 4) \).

---

### Question 4: Reflection Across the y-axis

Given points: \( T(6, -3) \) and \( U(3, 1) \).

Similarly, we reflect each point across the y-axis:
1. **Reflect Point T:**
\[
T' = (-6, -3)
\]

2. **Reflect Point U:**
\[
U' = (-3, 1)
\]

So the new points after reflection are \( T'(-6, -3) \) and \( U'(-3, 1) \).

---

### Question 5: Length of Line Segment After Reflection

Given endpoints: \( V(0, 5) \) and \( W(6, 5) \).

To reflect across the x-axis, we change the sign of the y-coordinate:
1. **Reflect Point V:**
\[
V' = (0, -5)
\]

2. **Reflect Point W:**
\[
W' = (6, -5)
\]

Now, we calculate the length of segment \( V'W' \):
\[
L = \sqrt{(6 - 0)^2 + (-5 - (-5))^2} = \sqrt{6^2 + 0} = \sqrt{36} = 6
\]

Thus, the length of the line segment remains 6 after reflection.

---

### Summary of Answers
1. Translated endpoints: \( (-5, 2) \) and \( (-3, -2) \)
2. Length after translation: \( 4 \)
3. Reflected endpoints across y-axis: \( (-1, 1) \) and \( (-4, 4) \)
4. New points after reflection across y-axis: \( (-6, -3) \) and \( (-3, 1) \)
5. Length after reflection across x-axis: \( 6 \)