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This guide explores the geometric transformations of reflection and rotation. We begin with the endpoints of line segment RS, followed by reflecting this segment across the y-axis. The guide then illustrates how to perform a 90-degree rotation about the origin after reflection. Various examples demonstrate the relationships and congruences of segments formed by these transformations. The principles outlined are supported by Theorem 9.5, revealing insights into the properties of parallel lines and perpendicularity in transformations, allowing for a better understanding of geometric relationships.
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The endpoints of RSare R(1, –3) and S(2, –6). Graph the image of RSafter the composition. STEP 1 Graph RS STEP 2 Reflect RSin the y-axis. R′S′ has endpoints R′(–1, –3) and S′(–2, –6). EXAMPLE 2 Find the image of a composition Reflection: in the y-axis Rotation: 90° about the origin SOLUTION
EXAMPLE 2 Find the image of a composition o Rotate R′S′ 90 about the origin. R′′S′′ has endpoints R′′(3, –1) andS′′(6, –2). STEP 3
In the diagram, a reflection in line kmaps GH to G′H′. A reflection in line m maps G′H′to G′′H′′. Also, HB = 9 andDH′′ = 4. a. Name any segments congruent to each segment: HG, HB, and GA ~ ~ ~ ~ a. HG H′G′ ,andHG H′′G′′. HB H′B. GA G′A . EXAMPLE 3 Use Theorem 9.5 SOLUTION
In the diagram, a reflection in line kmaps GH to G′H′. A reflection in line m maps G′H′to G′′H′′. Also, HB = 9 andDH′′ = 4. b. Yes, AC = BDbecause GG′′ and HH′′ are perpendicular to both kand m,so BDand ACare opposite sides of a rectangle. EXAMPLE 3 Use Theorem 9.5 b. Does AC = BD? Explain. SOLUTION
In the diagram, a reflection in line kmaps GH to G′H′. A reflection in line m maps G′H′to G′′H′′. Also, HB = 9 andDH′′ = 4. c.What is the length of GG′′ ? c. By the properties of reflections, H′B = 9 and H′D = 4. Theorem 9.5 implies that GG′′ = HH′′ = 2 BD, so the length of GG′′ is 2(9 + 4), or 26 units. EXAMPLE 3 Use Theorem 9.5 SOLUTION
3.Graph RSfrom Example 2. Do the rotation first, followed by the reflection. Does the order of the transformations matter? Explain. Yes; the resulting segment R′′ S ′′is not the same. for Examples 2 and 3 GUIDED PRACTICE SOLUTION
for Examples 2 and 3 GUIDED PRACTICE 4.In Example 3, part (c), explain how you know that GG′′ = HH′′. SOLUTION They are opposite sides of a parallelogram.
ANSWER 5. The preimage is reflected in line k , then in line m. Describe a single transformation that maps the blue figure to the green figure. Translation for Examples 2 and 3 GUIDED PRACTICE Use the figure below for Exercises 5 and 6. The distance between line k and line mis 1.6 centimeters.
6. What is the distance between Pand P′′? If you draw PP′ , what is its relationship with line k? Explain. ANSWER 3.2 cm; They are perpendicular. for Examples 2 and 3 GUIDED PRACTICE
