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Understanding Transformations: Translations and Reflections in Plane Geometry

This lesson delves into plane figures' transformations, focusing on translations and reflections. Students will learn to describe translations using vectors and find translation images through matrix and vector sums. The unit covers key concepts such as isometry, where all points of a figure are moved the same distance in a specified direction, and how to apply rules to characterize translations mathematically. Students will engage in practical examples and real-world applications, developing skills in both composition of transformations and understanding their geometric implications.

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Understanding Transformations: Translations and Reflections in Plane Geometry

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  1. Section 12-2 Transformations - Translations SPI 32D: determine whether the plane figure has been reflected given a diagram and vice versa • Objectives: • Describe translations using vectors • Find translation images using matrix and vector sums Unit Overview • Investigate Transformations • Reflections (this lesson) • Translations • Rotations • Composition of Transformations

  2. Translation (Slide) Isometry that maps all points of a figure the same distance in the same direction. The translation at the right maps VV’ and AA’ so that: AA’VV’ is a parallelogram if A, V, and V’ are noncollinear. AA’ = VV’ and AV = A’V’ if A, V, and V’ are collinear.

  3. Recall Matrices and Vectors Enter the Matrix x-coordinate y-coordinate • Vectors are written as: • <2, 3> • indicates direction of move for each point • Values correspond to (x, y)

  4. Write a Rule to Describe a Translation Write a rule to describe the translation PQRS → P’Q’R’S’ Take any point & its corresponding prime to find the vector. Use P(-1, -2) and its image P’(-5, -1) Find the horizontal change. -5 – (-1) = -4 Find the vertical change. -1 – (-2) = -1 The vector is: < -4, 1 > The Rule is: (x, y) → (x – 4, y + 1)

  5. Using Matrices to Find Images Use matrices to find the image of ∆MFH under the translation <4, -5>.

  6. Compositions A composition is a combination of two or more transformations. Each transformation is performed on the image of the preceding transformation. Example A game of chess where a knight is moved. The knight moves in a ‘L’ shape…. translate 3 units down and slide over 1 unit.

  7. Real-World: Using Compositions and Vectors Howard rides his bicycle 3 blocks east and 5 blocks North of a pharmacy to deliver a prescription. Then he rides 4 blocks west and 8 blocks south to make a second delivery. How many blocks is he now from the pharmacy? The vector 3, 5 represents a ride of 3 blocks east and 5 blocks north. The vector –4, –8 represents a ride of 4 blocks west and 8 blocks south Howard’s position after the second delivery is the sum of the vectors. 3, 5 + –4, –8 = 3 + (–4), 5 + (–8) = –1, –3, so Howard is 1 block south and 3 blocks west of the pharmacy.

  8. Do Now! Draw each figure on graph paper, and apply the given vector to draw the image. 1. 2.

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