Question

You are given the points left parenthesis 7 comma 5 right parenthesis​, left parenthesis negative 7 comma negative 5 right parenthesis​, left parenthesis negative 7 comma 5 right parenthesis and left parenthesis 7 comma negative 5 right parenthesis. What is the reflection of each point across the​ x-axis? Across the​ y-axis?
The four points are corners of a rectangle. Use pencil and paper. Tell how the points at opposite corners are related.
-10
-5
5
10
-10
-5
5
10
x
y



x y graph
Question content area bottom
Part 1
For the point ​(7​,5​), the reflection across the​ x-axis is
▼
left parenthesis negative 7 comma 5 right parenthesis .
left parenthesis 7 comma negative 5 right parenthesis .
left parenthesis negative 7 comma negative 5 right parenthesis .
The reflection across the​ y-axis is
▼
left parenthesis negative 7 comma negative 5 right parenthesis .
left parenthesis negative 7 comma 5 right parenthesis .
left parenthesis 7 comma negative 5 right parenthesis .

Answer 1

To find the reflections of each point across the x-axis and y-axis, we follow these rules:

1. **Reflection across the x-axis**: For a point \((x, y)\), the reflection across the x-axis is \((x, -y)\).
2. **Reflection across the y-axis**: For a point \((x, y)\), the reflection across the y-axis is \((-x, y)\).

Now, let's apply these rules to each of the given points:

1. **For the point \((7, 5)\)**:
- Reflection across the x-axis: \((7, -5)\)
- Reflection across the y-axis: \((-7, 5)\)

2. **For the point \((-7, -5)\)**:
- Reflection across the x-axis: \((-7, 5)\)
- Reflection across the y-axis: \((7, -5)\)

3. **For the point \((-7, 5)\)**:
- Reflection across the x-axis: \((-7, -5)\)
- Reflection across the y-axis: \((7, 5)\)

4. **For the point \((7, -5)\)**:
- Reflection across the x-axis: \((7, 5)\)
- Reflection across the y-axis: \((-7, -5)\)

### Summary of Reflections:

- For the point \((7, 5)\):
- Reflection across the x-axis: **(7, -5)**
- Reflection across the y-axis: **(-7, 5)**

- For the point \((-7, -5)\):
- Reflection across the x-axis: **(-7, 5)**
- Reflection across the y-axis: **(7, -5)**

- For the point \((-7, 5)\):
- Reflection across the x-axis: **(-7, -5)**
- Reflection across the y-axis: **(7, 5)**

- For the point \((7, -5)\):
- Reflection across the x-axis: **(7, 5)**
- Reflection across the y-axis: **(-7, -5)**

### Relationship Between Opposite Corners:

The relationship between the opposite corners of the rectangle can be described as follows:
- The opposite corners have coordinates that are negatives of each other across both axes.
- For example, the pairs:
- \((7, 5)\) and \((-7, -5)\) are opposite corners.
- \((7, -5)\) and \((-7, 5)\) are also opposite corners.
This means that each pair of opposite corners reflects across both axes to yield the other, affirming that they are symmetric around both axes.