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A 8- D. 7- 6- -5 4. C -3 -2 9 7 65 4 -2. e the apglet to ranslate quadrilateral ABCD 2 units lett o create quadrilateral A'B'C'D", then reflect A'B'C'D' over the line y=5 to create A"B"CD" WHat 中

A 8- D. 7- 6- -5 4. C -3 -2 9 7 65 4 -2. e the apglet to ranslate quadrilateral ABCD 2 units lett o create quadrilateral A'B'C'D", then reflect A'B'C'D' over the line y=5 to create A"B"CD" WHat 中

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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### Transformations of Quadrilateral ABCD In this interactive exercise, you'll use a graphing applet to explore geometric transformations. Follow the steps below to understand how translations and reflections can change the positions of points in a quadrilateral. #### Instructions: 1. **Initial Quadrilateral Position:** - Quadrilateral ABCD is plotted on a coordinate plane. - The vertices are labeled as follows: - \( A (4, 8) \) - \( B (6, 7) \) - \( C (3, 5) \) - \( D (2, 6) \) 2. **Translation:** - Translate quadrilateral ABCD 2 units to the left. - This will create a new quadrilateral A'B'C'D'. - Translation means shifting every point of the quadrilateral 2 units to the left without changing their relative positions. 3. **Reflection:** - Reflect quadrilateral A'B'C'D' over the line y = 5. - The reflection will flip the quadrilateral over the line \( y = 5 \), effectively creating a mirror image of A'B'C'D' to form A''B''C''D''. #### Question: **What are the coordinates of point D after performing these transformations?** To solve this, follow these steps: 1. **Translate D (2, 6)** 2 units to the left: - Subtract 2 from the x-coordinate: \( 2 - 2 = 0 \) - The new coordinates of D' are \( (0, 6) \). 2. **Reflect D' (0, 6)** over the line y = 5: - The line of reflection is horizontal at \( y = 5 \). - Reflecting over a horizontal line means the y-coordinate changes by the same amount to the other side of the line, while the x-coordinate remains the same. - Calculate the y-distance from the line: \( 6 - 5 = 1 \) - Reflect it to the other side: \( 5 - 1 = 4 \) - The new coordinates of D'' are \( (0, 4) \). Thus, the coordinates of point D after these transformations are \( (0, 4) \). Use the applet to perform these transformations and verify the coordinates. This
Transcribed Image Text:### Transformations of Quadrilateral ABCD In this interactive exercise, you'll use a graphing applet to explore geometric transformations. Follow the steps below to understand how translations and reflections can change the positions of points in a quadrilateral. #### Instructions: 1. **Initial Quadrilateral Position:** - Quadrilateral ABCD is plotted on a coordinate plane. - The vertices are labeled as follows: - \( A (4, 8) \) - \( B (6, 7) \) - \( C (3, 5) \) - \( D (2, 6) \) 2. **Translation:** - Translate quadrilateral ABCD 2 units to the left. - This will create a new quadrilateral A'B'C'D'. - Translation means shifting every point of the quadrilateral 2 units to the left without changing their relative positions. 3. **Reflection:** - Reflect quadrilateral A'B'C'D' over the line y = 5. - The reflection will flip the quadrilateral over the line \( y = 5 \), effectively creating a mirror image of A'B'C'D' to form A''B''C''D''. #### Question: **What are the coordinates of point D after performing these transformations?** To solve this, follow these steps: 1. **Translate D (2, 6)** 2 units to the left: - Subtract 2 from the x-coordinate: \( 2 - 2 = 0 \) - The new coordinates of D' are \( (0, 6) \). 2. **Reflect D' (0, 6)** over the line y = 5: - The line of reflection is horizontal at \( y = 5 \). - Reflecting over a horizontal line means the y-coordinate changes by the same amount to the other side of the line, while the x-coordinate remains the same. - Calculate the y-distance from the line: \( 6 - 5 = 1 \) - Reflect it to the other side: \( 5 - 1 = 4 \) - The new coordinates of D'' are \( (0, 4) \). Thus, the coordinates of point D after these transformations are \( (0, 4) \). Use the applet to perform these transformations and verify the coordinates. This
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