Math Problem Statement
Solution
From the image, it appears that the question involves a geometric transformation applied to the points , , and shown on the graph.
The selected answer is:
- "A 180° rotation about the origin, then a translation along the vector "
Let's break down this transformation:
-
180° Rotation about the Origin:
- A 180° rotation about the origin will change each point to .
- Applying this to the points:
- For point , if its original coordinates are , after rotation, it will become .
- Similarly, the points and will also be reflected over the origin to their respective new positions.
-
Translation along the vector :
- After rotating the points, this translation moves every point 2 units right and 2 units up.
- This means the new coordinates after rotation will be shifted accordingly.
The selected answer suggests this sequence of transformations results in the positions of , , and as shown.
Let me know if you'd like a more detailed explanation or if you have further questions!
Here are 5 related questions you might explore:
- How does a 90° clockwise rotation differ from a 180° rotation in terms of coordinate changes?
- What are the general rules for rotating points by 90°, 180°, and 270° about the origin?
- How do you determine the direction and magnitude of a vector used in translation?
- What is the effect of reflecting a point across the x-axis and y-axis on its coordinates?
- Can multiple transformations (rotation, translation, reflection) be combined into a single matrix?
Tip:
When performing a sequence of transformations, it's important to apply them in the specified order, as changing the order can lead to different results.
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Math Problem Analysis
Mathematical Concepts
Geometric Transformations
Coordinate Plane
Rotation
Translation
Reflection
Formulas
Rotation by 180°: (x, y) → (-x, -y)
Translation: (x, y) → (x + a, y + b)
Reflection in x-axis: (x, y) → (x, -y)
Theorems
Transformations in the coordinate plane
Rotation theorem
Suitable Grade Level
Grades 8-10