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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).

3 answers

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).

Write the coordinates of the reflected image.

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:

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