Problem Analysis:
The image shows two triangles:
- Triangle \( ABC \) in the upper part of the coordinate plane.
- Triangle \( A'B'C' \) in the lower part of the coordinate plane.
The task is to determine the transformation that maps triangle \( ABC \) to triangle \( A'B'C' \). This involves identifying whether the transformation is a reflection, rotation, translation, or some combination thereof.
Step-by-Step Solution:
#### 1.
Identify Key Points:
- Triangle \( ABC \):
- \( A = (3, 5) \)
- \( B = (4, 2) \)
- \( C = (2, 2) \)
- Triangle \( A'B'C' \):
- \( A' = (-3, -3) \)
- \( B' = (-4, -6) \)
- \( C' = (-2, -6) \)
#### 2.
Analyze the Transformation:
-
Translation: Check if there is a consistent shift in both \( x \)- and \( y \)-coordinates.
-
Reflection: Check if the points are mirrored across a line (e.g., the \( x \)-axis, \( y \)-axis, or the origin).
-
Rotation: Check if the points are rotated around a specific point.
#### 3.
Check for Reflection:
- Observe the coordinates of \( A \) and \( A' \):
- \( A = (3, 5) \)
- \( A' = (-3, -3) \)
- The \( x \)-coordinate changes sign, and the \( y \)-coordinate also changes sign. This suggests a reflection across the origin.
- Verify with other points:
- \( B = (4, 2) \) and \( B' = (-4, -6) \)
- \( C = (2, 2) \) and \( C' = (-2, -6) \)
- All points follow the pattern: \( (x, y) \rightarrow (-x, -y) \).
#### 4.
Conclusion:
- The transformation that maps triangle \( ABC \) to triangle \( A'B'C' \) is a
reflection across the origin.
Final Answer:
\[
\boxed{\text{Reflection across the origin}}
\]
Parent Tip: Review the logic above to help your child master the concept of sequence of transformations worksheet.
