Math Problem Statement
Solution
To solve this problem, we need to map to using rigid motions such as rotations and translations. We are given that the figures are congruent, meaning that the size and shape are identical, but their orientation and position may differ.
Step 1: Analyze the coordinates of the triangles
From the graph:
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Coordinates of :
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Coordinates of :
Step 2: Determine the rigid transformations
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Rotation: By observing the shapes and their positioning, it looks like needs a 90-degree counterclockwise rotation around the origin to match the orientation of .
- Coordinate rule for a 90-degree counterclockwise rotation: Applying this rule:
After this rotation, the triangle is still not in the same position as , so we need to follow up with a translation.
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Translation: We now translate the rotated triangle so that its vertices match . To do this, we can calculate the vector from a corresponding pair of points, say to .
- Translation vector:
Therefore, the coordinate notation for the translation is:
Final answer:
- Rotation: 90-degree counterclockwise rotation around the origin.
- Coordinate rule:
- Translation: By .
- Coordinate rule:
Let me know if you'd like further clarifications!
Here are 5 related questions that expand on this concept:
- How would the transformation rules change if the rotation was 90 degrees clockwise?
- Can you map to using reflections instead of rotations?
- What is the difference between a rigid transformation and a non-rigid transformation?
- How can you determine if two figures are congruent without performing the transformation?
- How would the transformation rules change if the rotation was around a point other than the origin?
Tip: When working with rotations, always pay attention to the direction (clockwise or counterclockwise) and the center of rotation.
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Math Problem Analysis
Mathematical Concepts
Congruence
Rotations
Translations
Coordinate Geometry
Formulas
Rotation formula: (x, y) → (-y, x)
Translation formula: (x, y) → (x - a, y + b)
Theorems
Congruent Figures Theorem
Rigid Transformation
Suitable Grade Level
Grades 8-10
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