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n maps triangle AE

n maps triangle AE

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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### Understanding Dilations on a Coordinate Plane #### Problem Statement: What dilation maps triangle ABC onto triangle A'B'C' below? #### Given Diagram: The diagram shows triangle ABC dilated onto triangle A'B'C' on a coordinate grid. The vertices of triangle ABC are labeled A, B, and C, and the vertices of the dilated triangle A'B'C' are labeled A', B', and C'. #### Options for Dilation Transformation: A. \((x, y) \rightarrow (2x, 2y)\) B. \((x, y) \rightarrow (0.5x, 0.5y)\) C. \((x, y) \rightarrow (3x, 3y)\) D. \((x, y) \rightarrow (-0.5x, -0.5y)\) Option B has been marked with a red "X," indicating it is the correct answer. #### Explanation of Diagram and Dilation: 1. **Triangle ABC (Original Triangle)**: - Plotted with vertices labeled A, B, and C on a standard coordinate grid. 2. **Triangle A'B'C' (Dilated Triangle)**: - The new vertices after dilation are labeled as A', B', and C'. - The diagram visually demonstrates how triangle ABC is transformed to produce triangle A'B'C'. #### Detailed Analysis: - **Option B**: The transformation \((x, y) \rightarrow (0.5x, 0.5y)\) indicates that each vertex of triangle ABC is reduced to half of its original coordinates, resulting in triangle A'B'C'. - When you apply this to any vertex (e.g., for point \(A(x, y)\)), the new coordinates would be \(A'(0.5x, 0.5y)\). - This matches the visual reduction in the size of triangle ABC in the diagram, confirming that the correct dilation factor is 0.5. The problem succinctly illustrates the concept of dilation in coordinate geometry using a clear visual aid and multiple-choice options, making it an excellent educational example.
Transcribed Image Text:### Understanding Dilations on a Coordinate Plane #### Problem Statement: What dilation maps triangle ABC onto triangle A'B'C' below? #### Given Diagram: The diagram shows triangle ABC dilated onto triangle A'B'C' on a coordinate grid. The vertices of triangle ABC are labeled A, B, and C, and the vertices of the dilated triangle A'B'C' are labeled A', B', and C'. #### Options for Dilation Transformation: A. \((x, y) \rightarrow (2x, 2y)\) B. \((x, y) \rightarrow (0.5x, 0.5y)\) C. \((x, y) \rightarrow (3x, 3y)\) D. \((x, y) \rightarrow (-0.5x, -0.5y)\) Option B has been marked with a red "X," indicating it is the correct answer. #### Explanation of Diagram and Dilation: 1. **Triangle ABC (Original Triangle)**: - Plotted with vertices labeled A, B, and C on a standard coordinate grid. 2. **Triangle A'B'C' (Dilated Triangle)**: - The new vertices after dilation are labeled as A', B', and C'. - The diagram visually demonstrates how triangle ABC is transformed to produce triangle A'B'C'. #### Detailed Analysis: - **Option B**: The transformation \((x, y) \rightarrow (0.5x, 0.5y)\) indicates that each vertex of triangle ABC is reduced to half of its original coordinates, resulting in triangle A'B'C'. - When you apply this to any vertex (e.g., for point \(A(x, y)\)), the new coordinates would be \(A'(0.5x, 0.5y)\). - This matches the visual reduction in the size of triangle ABC in the diagram, confirming that the correct dilation factor is 0.5. The problem succinctly illustrates the concept of dilation in coordinate geometry using a clear visual aid and multiple-choice options, making it an excellent educational example.
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