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Give the coordinates of ABC. Reflect over the line horizontal line y = -2.Then give the coordinates of A'B'C' 5 A B C

Give the coordinates of ABC. Reflect over the line horizontal line y = -2.Then give the coordinates of A'B'C' 5 A B C

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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### Reflection Over a Horizontal Line **Objective:** - Find the coordinates of triangle ABC. - Reflect triangle ABC over the horizontal line \( y = -2 \). - Determine the coordinates of the reflected triangle A'B'C'. #### Original Coordinates of Triangle ABC - **Point A:** (2, 5) - **Point B:** (3, 2) - **Point C:** (6, 3) #### Reflection Process To reflect a point over a horizontal line \( y = k \), keep the \( x \)-coordinate the same and use the formula for the new \( y \)-coordinate: \[ y' = 2k - y \] For reflection over \( y = -2 \): - **Reflection of Point A (2, 5):** \[ y' = 2(-2) - 5 = -4 - 5 = -9 \quad \Rightarrow \quad A'(2, -9) \] - **Reflection of Point B (3, 2):** \[ y' = 2(-2) - 2 = -4 - 2 = -6 \quad \Rightarrow \quad B'(3, -6) \] - **Reflection of Point C (6, 3):** \[ y' = 2(-2) - 3 = -4 - 3 = -7 \quad \Rightarrow \quad C'(6, -7) \] #### Coordinates of Reflected Triangle A'B'C' - **Point A':** (2, -9) - **Point B':** (3, -6) - **Point C':** (6, -7) ### Diagram Explanation The graph provides a visual representation of triangle ABC in the first quadrant of a Cartesian plane. It is reflected over the horizontal line \( y = -2 \) to form triangle A'B'C' in the lower quadrants. The reflection maintains the shapes and relative distances of the points while changing their \( y \)-coordinates as described.
Transcribed Image Text:### Reflection Over a Horizontal Line **Objective:** - Find the coordinates of triangle ABC. - Reflect triangle ABC over the horizontal line \( y = -2 \). - Determine the coordinates of the reflected triangle A'B'C'. #### Original Coordinates of Triangle ABC - **Point A:** (2, 5) - **Point B:** (3, 2) - **Point C:** (6, 3) #### Reflection Process To reflect a point over a horizontal line \( y = k \), keep the \( x \)-coordinate the same and use the formula for the new \( y \)-coordinate: \[ y' = 2k - y \] For reflection over \( y = -2 \): - **Reflection of Point A (2, 5):** \[ y' = 2(-2) - 5 = -4 - 5 = -9 \quad \Rightarrow \quad A'(2, -9) \] - **Reflection of Point B (3, 2):** \[ y' = 2(-2) - 2 = -4 - 2 = -6 \quad \Rightarrow \quad B'(3, -6) \] - **Reflection of Point C (6, 3):** \[ y' = 2(-2) - 3 = -4 - 3 = -7 \quad \Rightarrow \quad C'(6, -7) \] #### Coordinates of Reflected Triangle A'B'C' - **Point A':** (2, -9) - **Point B':** (3, -6) - **Point C':** (6, -7) ### Diagram Explanation The graph provides a visual representation of triangle ABC in the first quadrant of a Cartesian plane. It is reflected over the horizontal line \( y = -2 \) to form triangle A'B'C' in the lower quadrants. The reflection maintains the shapes and relative distances of the points while changing their \( y \)-coordinates as described.
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