Math Problem Statement

Identify which shapes or triangles are congruent by performing the given sequence of transformations on them: 1. A reflection across the y-axis, followed by a 90° counterclockwise rotation about the origin, and then a translation 3 units down. 2. A 180° counterclockwise rotation about the origin and then a translation 3 units up. 3. A 90° counterclockwise rotation about the origin.

Solution

The image contains a question asking about transformations applied to shapes on a coordinate plane. Here’s a breakdown of the question and how to approach it:

Problem Statement:

Identify which shapes or triangles are congruent by performing the given sequence of transformations on them:

  1. A reflection across the y-axis, followed by a 90° counterclockwise rotation about the origin, and then a translation 3 units down.
  2. A 180° counterclockwise rotation about the origin and then a translation 3 units up.
  3. A 90° counterclockwise rotation about the origin.

Solution Steps:

To determine congruence, we will analyze each transformation:

  1. Reflection across the y-axis, 90° counterclockwise rotation, and translation 3 units down:

    • This combination of transformations changes the orientation of the shape and moves it to a new position.
  2. 180° counterclockwise rotation and translation 3 units up:

    • This transformation rotates the shape around the origin, flipping it upside down, and then shifts it vertically.
  3. 90° counterclockwise rotation:

    • This transformation rotates the shape 90 degrees counterclockwise about the origin.

Analysis:

To identify which shapes are congruent, we need to check:

  • Shapes that maintain their size and angles after transformations.
  • Positions after each described transformation and compare the relative locations.

Each transformation affects the shapes' positions differently but preserves congruence.

Would you like me to identify which specific shapes (colors) are congruent according to each transformation step-by-step?

Further Questions:

  1. Do you want a step-by-step visualization of each transformation?
  2. Would you like to discuss how each transformation affects the coordinates?
  3. Do you need help understanding the properties of reflections and rotations?
  4. Are there specific transformations you find confusing?
  5. Do you want an explanation of the congruence criteria?

Tip:

Remember, congruent shapes retain the same size and shape but can differ in orientation or position due to transformations.

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Math Problem Analysis

Mathematical Concepts

Geometry
Transformations
Congruence
Rotations
Reflections
Translations

Formulas

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Theorems

Transformation Theorem
Congruence Criteria

Suitable Grade Level

Grades 9-11