504 CHAPTER 8 Geometry In Example 2, note that other than the vertex labels, the positions of the hexagon before and after the reflection are identical. We will revisit reflections like this when we discuss reflective symmetry later in this section. Now consider hexagon ABCDEF in Fig. 8.53 and its reflection about line m, hexagon A BC D E F ′ ′ ′ ′ ′ ′ in Fig. 8.54. Notice that the positions of the hexagon before and after the reflection, relative to line m, are not the same. If we line up reflection line m in Fig. 8.53 and Fig. 8.54, we see that hexagon ABCDEF and hexagon A BC D E F ′ ′ ′ ′ ′ ′ are in different positions. We next see that vertex B in hexagon ABCDEF is 2 units to the right of l. Thus, vertex B′ in the reflected hexagon will be 2 units to the left of l. Notice that vertex B′ of the reflected hexagon is in the same location as vertex A of hexagon ABCDEF. We continue this process to determine the locations of vertices C D E , , , ′ ′ ′ and F′ of the reflected hexagon. Notice that each vertex of the reflected hexagon is in the same location as a vertex of hexagon ABCDEF. Finally, we draw the line segments to complete the reflected hexagon A BC D E F ′ ′ ′ ′ ′ ′ (see Fig. 8.52). 7 Now try Exercise 13 Profiles in Mathematics NASA’s Hidden Figures m Katherine Johnson The 2016 award-winning movie Hidden Figures is a biographical drama based upon Margot Lee Shetterly’s book Hidden Figures: The American Dream and the Untold Story of the Black Women Mathematicians Who Helped Win the Space Race . The movie focuses on Katherine Johnson (1918–2020) (BA in Mathematics, West Virginia State College), Dorothy Vaughan (1910–2008) (BA in Mathematics, Wilberforce University), and Mary Jackson (1921–2005) (BS in Mathematics and Physical Science, Hampton Institute), who made many critical advancements at NASA during the early years of the space program. The movie was nominated for three Academy Awards, including Best Motion Picture. Katherine Johnson’s expertise included analytic geometry and calculating flight trajectories. In 2015, President Obama awarded the Presidential Medal of Freedom to Johnson. In 2016, NASA’s Johnson Computational Research Facility was named in her honor. Dorothy Vaughan was the unofficial supervisor of the African American group of women who worked for NASA. She taught herself programming and was promoted to head of the computer programming department. Mary Jackson worked on the heat shield of the space capsule that carried John Glenn into space. She became the first African American woman NASA engineer. Because of segregation, she had to overcome many hurdles to attend an all-white school where she could take advanced engineering courses. These and many other African American women mathematicians known as “human computers” made many contributions to the space program and played a huge role in many of NASA’s achievements. A B D E F C m Figure 8.53 B9 A9 E9 D9 C9 F9 m Figure 8.54 Translations The next rigid motion we will discuss is the translation . In a translation, we simply move a figure along a straight line to a new position. Definition: Translation or Glide A translation, or glide , is a rigid motion that moves a geometric figure by sliding it along a straight line segment in the plane. The direction and length of the line segment completely determine the translation. After conducting a translation, we say the figure was translated to a new position. A concise way to indicate the direction and the distance that a figure is moved during a translation is with a translation vector . In mathematics, vectors are typically represented with boldface letters. For example, in Fig. 8.55 we see trapezoid ABCD and a translation vector, v , which is pointing to the right and upward. This translation vector indicates a translation of 9 units to the right and 4 units upward. Notice that in Fig. 8.55 the translated vector appears on the right side of the polygon. The placement of the translation vector does not matter. Science History Images/Photo Researchers/Alamy Stock Photo
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