Math Problem Statement

Show how each vertex of triangle EFG maps to its image using a reflection across the x-axis. Complete the table with the coordinates of the image points E', F', and G'.

Solution

The task involves reflecting the triangle ΔEFG\Delta EFG to its image ΔEFG\Delta E'F'G' over the x-axis. You need to determine the coordinates of the reflected image points E,F,GE', F', G', which correspond to points E,F,GE, F, G after reflection.

Coordinates of ΔEFG\Delta EFG:

  • E(3,5)E (-3, 5)
  • F(2,2)F (-2, 2)
  • G(4,2)G (-4, 2)

Reflection rule:

When reflecting over the x-axis, the y-coordinate of each point changes sign, while the x-coordinate remains the same. The reflection formula is: (x,y)(x,y)(x, y) \to (x, -y)

Applying the reflection:

  • For point E(3,5)E (-3, 5), the reflection is E(3,5)E' (-3, -5).
  • For point F(2,2)F (-2, 2), the reflection is F(2,2)F' (-2, -2).
  • For point G(4,2)G (-4, 2), the reflection is G(4,2)G' (-4, -2).

Completed table:

ΔEFG\Delta EFGΔEFG\Delta E'F'G'
E(3,5)E (-3, 5)E(3,5)E' (-3, -5)
F(2,2)F (-2, 2)F(2,2)F' (-2, -2)
G(4,2)G (-4, 2)G(4,2)G' (-4, -2)

Explanation in words:

To map ΔEFG\Delta EFG to ΔEFG\Delta E'F'G', you reflect each vertex over the x-axis. This means for every point, the x-coordinate stays the same, while the y-coordinate changes sign. For example, point E(3,5)E (-3, 5) is reflected to E(3,5)E' (-3, -5), and the same rule applies to the other points.

Would you like more details or have any questions?

Additional questions:

  1. What happens if you reflect ΔEFG\Delta EFG over the y-axis instead of the x-axis?
  2. How would the coordinates change if you reflected the triangle over both the x- and y-axes?
  3. What is the distance between point EE and its reflection EE'?
  4. How can you use the reflection formula to determine if two points are symmetric with respect to the x-axis?
  5. Can you describe the transformation if the triangle is reflected over a diagonal line like y=xy = x?

Tip:

Reflections maintain the size and shape of a figure, but they reverse the orientation relative to the axis of reflection.

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Math Problem Analysis

Mathematical Concepts

Geometry
Reflections
Coordinate Plane

Formulas

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

Theorems

Reflection Theorem

Suitable Grade Level

Grades 8-10