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3. Part B. Jim is confused now because he knows that he learned that rigid transformations preserve distance. Yet, he says that he can see that the distances AC and BD in the completed figure are not the same! Explain the error in Jim's thinking. Write your answer in the space provided. BI

3. Part B. Jim is confused now because he knows that he learned that rigid transformations preserve distance. Yet, he says that he can see that the distances AC and BD in the completed figure are not the same! Explain the error in Jim's thinking. Write your answer in the space provided. BI

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 Rigid Transformations in Geometry **Directions: Read the passage and answer the question(s) that follow.** A rigid transformation, \( J \), of the plane takes a point, \( A \), as input and gives \( C \) as output (i.e., \( J(A) = C \)). Similarly, \( J \)(B) = D for input point \( B \) and output point \( D \). Since rigid transformations preserve distance, \( AC = CD \). --- ### Part B **Problem Statement:** Jim is confused now because he knows that he learned that rigid transformations preserve distance. Yet, he says that he can see that the distances \( AC \) and \( BD \) in the completed figure are not the same! Explain the error in Jim's thinking. **Answer:** The rigid transformation would only be congruent in the completed figure if it were a translation. --- ### Explanation In the provided text, we are exploring rigid transformations of the plane. A rigid transformation is a way of moving a figure where the shape and size do not alter. These transformations include translations, rotations, and reflections, all of which preserve distances between points. In this scenario: - \( J \) is a rigid transformation. - \( J(A) = C \) means point \( A \) is mapped to point \( C \) under the transformation. - \( J(B) = D \) means point \( B \) is mapped to point \( D \) under the same transformation. **Key Concept:** Any rigid transformation preserves the distance between points. Therefore, if you measure the length of \( AC \) after the transformation, it should be equal to the length of \( CD \). **Jim's Error:** Jim believes that the distances between points \( AC \) and \( BD \) are not the same. However, according to the property of rigid transformations, these distances should be congruent if they are part of the same transformed figure. Jim's confusion might arise from misinterpreting other distances that are not directly comparable within the transform. If the transformation is a simple translation, the distances should match directly because every point moves the same distance in the same direction. However, for other rigid transformations like rotations or reflections, Jim needs to consider how these operations also preserve distance, but not necessarily align them in a straightforward way.
Transcribed Image Text:### Understanding Rigid Transformations in Geometry **Directions: Read the passage and answer the question(s) that follow.** A rigid transformation, \( J \), of the plane takes a point, \( A \), as input and gives \( C \) as output (i.e., \( J(A) = C \)). Similarly, \( J \)(B) = D for input point \( B \) and output point \( D \). Since rigid transformations preserve distance, \( AC = CD \). --- ### Part B **Problem Statement:** Jim is confused now because he knows that he learned that rigid transformations preserve distance. Yet, he says that he can see that the distances \( AC \) and \( BD \) in the completed figure are not the same! Explain the error in Jim's thinking. **Answer:** The rigid transformation would only be congruent in the completed figure if it were a translation. --- ### Explanation In the provided text, we are exploring rigid transformations of the plane. A rigid transformation is a way of moving a figure where the shape and size do not alter. These transformations include translations, rotations, and reflections, all of which preserve distances between points. In this scenario: - \( J \) is a rigid transformation. - \( J(A) = C \) means point \( A \) is mapped to point \( C \) under the transformation. - \( J(B) = D \) means point \( B \) is mapped to point \( D \) under the same transformation. **Key Concept:** Any rigid transformation preserves the distance between points. Therefore, if you measure the length of \( AC \) after the transformation, it should be equal to the length of \( CD \). **Jim's Error:** Jim believes that the distances between points \( AC \) and \( BD \) are not the same. However, according to the property of rigid transformations, these distances should be congruent if they are part of the same transformed figure. Jim's confusion might arise from misinterpreting other distances that are not directly comparable within the transform. If the transformation is a simple translation, the distances should match directly because every point moves the same distance in the same direction. However, for other rigid transformations like rotations or reflections, Jim needs to consider how these operations also preserve distance, but not necessarily align them in a straightforward way.
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