Math Problem Statement
Solution
The problem in the image asks for the image of the point after two transformations:
- Translation by , which seems to refer to a vector translation of .
- Rotation by counterclockwise about the origin .
Step 1: Translation
We first translate point by the vector .
- New coordinates of :
Step 2: Rotation
Next, we rotate the point counterclockwise by around the origin . The transformation formula for a counterclockwise rotation is:
- New coordinates of after rotation: Applying this to :
Final Answer:
The image of the point after translation and rotation is .
Would you like further details or explanations? Here are some related questions to practice:
- How would the result change if the rotation was clockwise?
- What is the general formula for rotating a point by ?
- How do you apply a translation using different vectors?
- What are the coordinates of the point if the rotation was around a different center?
- What happens if the translation vector is reversed?
Tip: Always apply translation first, followed by rotation, unless the problem specifies otherwise.
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Math Problem Analysis
Mathematical Concepts
Translation
Rotation
Coordinate Geometry
Formulas
Translation formula: (x', y') = (x + dx, y + dy)
Rotation formula for 90 degrees counterclockwise: (x', y') = (-y, x)
Theorems
Rotation Theorem
Suitable Grade Level
Grades 8-10
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