Math Problem Statement

The graph shows pentagons PQRST and P'Q'R'S'T'. Which sequence of transformations maps PQRST onto P'Q'R'S'T'?

Solution

Let's analyze the transformations that map pentagon PQRST onto P'Q'R'S'T'.

Step 1: Identify the Relationship Between PQRST and P'Q'R'S'T'

  • The original pentagon PQRST is in the first quadrant.
  • The transformed pentagon P'Q'R'S'T' is in the second quadrant, which suggests a reflection or rotation.

Step 2: Test Possible Transformations

Option 1: Reflection Across the y-axis Followed by a Translation Right 4 Units and Up 3 Units

  • A reflection over the y-axis would place the shape in the second quadrant, which matches the new location.
  • However, reflecting the shape over the y-axis flips the orientation, but does not align it properly.
  • A translation of right 4, up 3 does not fully match the new pentagon.

This option seems incorrect.

Option 2: Translation Left 4 Units and Down 7 Units, then Rotation 180°

  • Moving left 4 units and down 7 units shifts the shape, but the orientation remains the same.
  • A 180° rotation would correctly align it by flipping it around the origin.

This option appears to be correct!

Option 3: Rotation 90° Clockwise, then Translation Left 4 Units and Up 12 Units

  • A 90° clockwise rotation would place the shape in a different orientation.
  • Translation left 4, up 12 would not correctly align the figure.

This option is incorrect.

Conclusion:

The correct answer is "A translation left 4 units and down 7 units followed by a rotation 180° around the origin."

Would you like a step-by-step coordinate transformation breakdown?


5 Related Questions:

  1. What happens to coordinates when reflected across the y-axis?
  2. How do you determine the correct transformation sequence for shapes?
  3. What is the effect of a 180° rotation on a shape?
  4. How can you verify transformations using coordinate points?
  5. What transformations preserve shape size and orientation?

Tip:

When analyzing transformations, always check orientation, position, and size before selecting an answer!

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Math Problem Analysis

Mathematical Concepts

Geometric Transformations
Coordinate Geometry

Formulas

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Theorems

Reflection across the y-axis
Rotation 180°
Translation of points

Suitable Grade Level

Grade 8-9