Problem Analysis:
The image shows a transformation of rotation applied to a geometric shape. The task is to analyze the transformation and explain how it was performed.
#### Key Observations:
1.
Original Shape (Orange Triangle):
- The orange triangle is located in the second quadrant.
- Its vertices are approximately at:
- \( A(-3, 4) \)
- \( B(-2, 1) \)
- \( C(-5, 1) \)
2.
Transformed Shape (Blue Triangle):
- The blue triangle is located in the fourth quadrant.
- Its vertices are approximately at:
- \( A'(3, -4) \)
- \( B'(2, -1) \)
- \( C'(5, -1) \)
3.
Transformation Type:
- The transformation appears to be a
rotation because the shape maintains its size and orientation relative to the axes but changes its position.
4.
Rotation Center:
- The rotation seems to be centered around the origin \((0, 0)\), as the coordinates of the vertices change sign and their relative positions suggest a symmetry about the origin.
---
Step-by-Step Solution:
#### 1.
Identify the Rotation:
- The transformation from the orange triangle to the blue triangle involves flipping the coordinates across both axes. This suggests a
180° rotation about the origin.
- In a 180° rotation about the origin:
- A point \((x, y)\) transforms to \((-x, -y)\).
#### 2.
Verify the Transformation:
-
Vertex \( A(-3, 4) \):
- After a 180° rotation: \((-(-3), -(4)) = (3, -4)\).
- This matches the coordinates of \( A'(3, -4) \).
-
Vertex \( B(-2, 1) \):
- After a 180° rotation: \((-(-2), -(1)) = (2, -1)\).
- This matches the coordinates of \( B'(2, -1) \).
-
Vertex \( C(-5, 1) \):
- After a 180° rotation: \((-(-5), -(1)) = (5, -1)\).
- This matches the coordinates of \( C'(5, -1) \).
#### 3.
Conclusion:
- All vertices of the orange triangle transform correctly to the corresponding vertices of the blue triangle under a 180° rotation about the origin.
---
Final Answer:
The transformation applied is a
180° rotation about the origin.
\[
\boxed{180^\circ \text{ rotation about the origin}}
\]
Parent Tip: Review the logic above to help your child master the concept of mathematical transformations.
