Math Problem Statement

Which figure shows a reflection of pre-image DEFG over the x-axis? Responses

Two trapezoids graphed on a grid. Trapezoid D E F G is in Quadrant One and has coordinates D (one, one), E (two, three), F (three, three), and G (four, one). Trapezoid D-prime, E-prime, F-prime, G-prime is in Quadrant Three and has coordinates D-prime (negative one, negative one), E-prime (negative two, negative three), F-prime (negative three, negative three), and G-prime (negative four, negative one). Image with alt text: Two trapezoids graphed on a grid. Trapezoid D E F G is in Quadrant One and has coordinates D (one, one), E (two, three), F (three, three), and G (four, one). Trapezoid D-prime, E-prime, F-prime, G-prime is in Quadrant Three and has coordinates D-prime (negative one, negative one), E-prime (negative two, negative three), F-prime (negative three, negative three), and G-prime (negative four, negative one).

Two trapezoids graphed on a grid. Trapezoid D E F G is in Quadrant One and has coordinates D (one, one), E (two, three), F (three, three), and G (four, one). Trapezoid D-prime, E-prime, F-prime, G-prime is in Quadrant Two and has coordinates D-prime (negative one, one), E-prime (negative two, three), F-prime (negative three, three), and G-prime (negative four, one). Image with alt text: Two trapezoids graphed on a grid. Trapezoid D E F G is in Quadrant One and has coordinates D (one, one), E (two, three), F (three, three), and G (four, one). Trapezoid D-prime, E-prime, F-prime, G-prime is in Quadrant Two and has coordinates D-prime (negative one, one), E-prime (negative two, three), F-prime (negative three, three), and G-prime (negative four, one).

Two trapezoids graphed on a grid. Trapezoid D E F G is in Quadrant One and has coordinates D (one, one), E (two, three), F (three, three), and G (four, one). Trapezoid D-prime, E-prime, F-prime, G-prime is in Quadrant Four and has coordinates D-prime (one, negative one), E-prime (two, negative three), F-prime (three, negative three), and G-prime (four, negative one). Image with alt text: Two trapezoids graphed on a grid. Trapezoid D E F G is in Quadrant One and has coordinates D (one, one), E (two, three), F (three, three), and G (four, one). Trapezoid D-prime, E-prime, F-prime, G-prime is in Quadrant Four and has coordinates D-prime (one, negative one), E-prime (two, negative three), F-prime (three, negative three), and G-prime (four, negative one).

Solution

To find the correct reflection of trapezoid DEFG over the x-axis, we need to apply the rule of reflection across the x-axis. When a point (x,y)(x, y) is reflected over the x-axis, its image will be (x,−y)(x, -y).

Let's check each option:

  1. First option (in Quadrant Three):

    • The original coordinates are:
      • D(1,1)→D′(−1,−1)D(1, 1) \to D'(-1, -1)
      • E(2,3)→E′(−2,−3)E(2, 3) \to E'(-2, -3)
      • F(3,3)→F′(−3,−3)F(3, 3) \to F'(-3, -3)
      • G(4,1)→G′(−4,−1)G(4, 1) \to G'(-4, -1)
    • This transformation is a reflection over the x-axis because the yy-coordinates are inverted while the xx-coordinates stay the same.

  2. Second option (in Quadrant Two):

    • The original coordinates are:
      • D(1,1)→D′(−1,1)D(1, 1) \to D'(-1, 1)
      • E(2,3)→E′(−2,3)E(2, 3) \to E'(-2, 3)
      • F(3,3)→F′(−3,3)F(3, 3) \to F'(-3, 3)
      • G(4,1)→G′(−4,1)G(4, 1) \to G'(-4, 1)
    • This is not a reflection over the x-axis, as the yy-coordinates are not inverted.

  3. Third option (in Quadrant Four):

    • The original coordinates are:
      • D(1,1)→D′(1,−1)D(1, 1) \to D'(1, -1)
      • E(2,3)→E′(2,−3)E(2, 3) \to E'(2, -3)
      • F(3,3)→F′(3,−3)F(3, 3) \to F'(3, -3)
      • G(4,1)→G′(4,−1)G(4, 1) \to G'(4, -1)
    • This correctly reflects the trapezoid over the x-axis.

So, the third option shows the reflection of the pre-image DEFG over the x-axis.

Would you like further details or explanations?

Here are 5 related questions:

  1. How do you reflect a figure over the y-axis?
  2. What happens if you reflect a point across the line y=xy = x?
  3. How do reflections affect the area or shape of the figure?
  4. Can a reflection change the orientation of a figure?
  5. What are the other types of transformations (like translations and rotations) that can be applied to a shape?

Tip: Always check the coordinates carefully when reflecting over the x-axis or y-axis to ensure that only the relevant coordinate is changed (x stays the same for x-axis reflection, and y stays the same for y-axis reflection).

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Math Problem Analysis

Mathematical Concepts

Geometry
Reflections
Coordinate Geometry

Formulas

Reflection over the x-axis: (x, y) → (x, -y)

Theorems

Reflection theorem over the x-axis

Suitable Grade Level

Grades 6-8

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