Math Problem Statement

Graph the image of ΔGHI after the following sequence of transformations: Reflection across the line y = 1, Rotation 180° counterclockwise around the origin.

Solution

To solve this problem, we will break it into two transformations as specified:

  1. Reflection across the line y=1y = 1.
  2. Rotation 180° counterclockwise around the origin.

Step 1: Reflection across the line y=1y = 1

We need to reflect the coordinates of points GG, HH, and II across the horizontal line y=1y = 1.

  • The formula for reflecting a point (x,y)(x, y) across a horizontal line y=ky = k is: (x,y)=(x,2ky)(x', y') = (x, 2k - y) In this case, k=1k = 1.

Step 2: 180° Rotation Counterclockwise Around the Origin

After reflection, we will rotate the new points 180° counterclockwise around the origin. The formula for rotating a point (x,y)(x, y) by 180° counterclockwise is: (x,y)=(x,y)(x', y') = (-x, -y)

Let me now apply these transformations to the triangle's points and plot the resulting image.

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Math Problem Analysis

Mathematical Concepts

Geometric Transformations
Reflections
Rotations

Formulas

Reflection formula across a horizontal line y = k: (x', y') = (x, 2k - y)
Rotation formula for 180° counterclockwise: (x', y') = (-x, -y)

Theorems

Geometric Transformation Theorems

Suitable Grade Level

Grades 8-10