Question

The graph of ∆DEF
has vertices D(1
, 1)
, E(–3
, –1)
, and F(4
, –1)
.

Select the graph of ∆DEF
and its image after a reflection across the line y=2
. The line of reflection is included.

Multiple choice question.

A horizontal line at y equals 2. Triangles D E F and D prime E prime F prime. D prime (negative 1, 3), E prime (4, 5), F prime (negative 3, 5).


A horizontal line at y equals 2. Triangles D E F and D prime E prime F prime. D prime (negative 1, 1), E prime (4, 3), F prime (negative 3, 3).


A horizontal line at y equals 2. Triangles D E F and D prime E prime F prime. D prime (1, 1), E prime (negative 3, 3), F prime (4, 3).


A horizontal line at y equals 2. Triangles D E F and D prime E prime F prime. D prime (1, 3), E prime (negative 3, 5), F prime (4, 5).

Answer 1

Write the coordinates of the reflected image.

Answer 2

To find the image of triangle \( \Delta DEF \) after a reflection across the line \( y = 2 \), we can follow these steps:

1. **Identify the vertices of triangle DEF:**
- \( D(1, 1) \)
- \( E(-3, -1) \)
- \( F(4, -1) \)

2. **Reflection across the line \( y = 2 \):**
- The reflection rule states that if a point \((x, y)\) is reflected across a horizontal line \( y = k \), we can find the new point as follows:
- The distance from the point to the line is \( |y - k| \).
- The y-coordinate of the reflected point will be determined by moving the same distance above the line.

Given \( y = 2\):
- **Reflect point D(1, 1)**:
- Distance to line: \( |1 - 2| = 1 \)
- Reflected point \( D' = (1, 2 + 1) = (1, 3) \)

- **Reflect point E(-3, -1)**:
- Distance to line: \( |-1 - 2| = 3 \)
- Reflected point \( E' = (-3, 2 + 3) = (-3, 5) \)

- **Reflect point F(4, -1)**:
- Distance to line: \( |-1 - 2| = 3 \)
- Reflected point \( F' = (4, 2 + 3) = (4, 5) \)

3. **Reflected vertices:**
- \( D'(1, 3) \)
- \( E'(-3, 5) \)
- \( F'(4, 5) \)

Now, let's look at the provided options:

1. **Option 1:** D'(−1, 3), E'(4, 5), F'(−3, 5)
2. **Option 2:** D'(−1, 1), E'(4, 3), F'(−3, 3)
3. **Option 3:** D'(1, 1), E'(−3, 3), F'(4, 3)
4. **Option 4:** D'(1, 3), E'(−3, 5), F'(4, 5)

**Correct Option:**
The correct reflection points are:

- \( D'(1, 3) \)
- \( E'(-3, 5) \)
- \( F'(4, 5) \)

So the correct answer is **Option 4**: D'(1, 3), E'(-3, 5), F'(4, 5).

Answer 3

The coordinates of the reflected image of triangle \( \Delta DEF \) after reflecting across the line \( y = 2 \) are:

- **D'**: \( (1, 3) \)
- **E'**: \( (-3, 5) \)
- **F'**: \( (4, 5) \)

So, the coordinates of the reflected image are:

- \( D' (1, 3) \)
- \( E' (-3, 5) \)
- \( F' (4, 5) \)