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Transformations in Coordinate Geometry Transformations in the Coordinate Plane

Transformations in Coordinate Geometry Transformations in the Coordinate Plane. Holt Geometry.

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Transformations in Coordinate Geometry Transformations in the Coordinate Plane

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  1. Transformations in Coordinate GeometryTransformations in the Coordinate Plane Holt Geometry

  2. A transformationis a change in the position, size, or shape of a figure. The original figure is called the preimage. The resulting figure is called the image. A transformation maps the preimage to the image. Arrow notation () is used to describe a transformation, and primes (’) are used to label the image.

  3. Example 1A: Identifying Transformation Identify the transformation. Then use arrow notation to describe the transformation. The transformation cannot be a reflection because each point and its image are not the same distance from a line of reflection. 90° rotation, ∆ABC ∆A’B’C’

  4. Example 1B: Identifying Transformation Identify the transformation. Then use arrow notation to describe the transformation. The transformation cannot be a translation because each point and its image are not in the same relative position. reflection, DEFG D’E’F’G’

  5. Check It Out! Example 1 Identify each transformation. Then use arrow notation to describe the transformation. a. b. translation; MNOP M’N’O’P’ rotation; ∆XYZ ∆X’Y’Z’

  6. Example 2: Drawing and Identifying Transformations A figure has vertices at A(1, –1), B(2, 3), and C(4, –2). After a transformation, the image of the figure has vertices at A'(–1, –1), B'(–2, 3), and C'(–4, –2). Draw the preimage and image. Then identify the transformation. Plot the points. Then use a straightedge to connect the vertices. The transformation is a reflection across the y-axis because each point and its image are the same distance from the y-axis.

  7. Check It Out! Example 2 A figure has vertices at E(2, 0), F(2, -1), G(5, -1), and H(5, 0). After a transformation, the image of the figure has vertices at E’(0, 2), F’(1, 2), G’(1, 5), and H’(0, 5). Draw the preimage and image. Then identify the transformation. Plot the points. Then use a straightedge to connect the vertices. The transformation is a 90° counterclockwise rotation.

  8. To find coordinates for the image of a figure in a translation, add a to the x-coordinates of the preimage and add b to the y-coordinates of the preimage. Translations can also be described by a rule such as (x, y)  (x + a, y + b).

  9. Example 3: Translations in the Coordinate Plane Find the coordinates for the image of ∆ABC after the translation (x, y)  (x + 2, y - 1). Draw the image. Step 1 Find the coordinates of ∆ABC. The vertices of ∆ABC are A(–4, 2), B(–3, 4), C(–1, 1).

  10. Example 3 Continued Step 2 Apply the rule to find the vertices of the image. A’(–4 + 2, 2 – 1) = A’(–2, 1) B’(–3 + 2, 4 – 1) = B’(–1, 3) C’(–1 + 2, 1 – 1) = C’(1, 0) Step 3 Plot the points. Then finish drawing the image by using a straightedge to connect the vertices.

  11. Check It Out! Example 3 Find the coordinates for the image of JKLM after the translation (x, y)  (x – 2, y + 4). Draw the image. Step 1 Find the coordinates of JKLM. The vertices of JKLM are J(1, 1), K(3, 1), L(3, –4), M(1, –4), .

  12. J’ K’ M’ L’ Check It Out! Example 3 Continued Step 2 Apply the rule to find the vertices of the image. J’(1 – 2, 1 + 4) = J’(–1, 5) K’(3 – 2, 1 + 4) = K’(1, 5) L’(3 – 2, –4 + 4) = L’(1, 0) M’(1 – 2, –4 + 4) = M’(–1, 0) Step 3 Plot the points. Then finish drawing the image by using a straightedge to connect the vertices.

  13. Step 1 Choose two points. Choose a Point A on the preimage and a corresponding Point A’ on the image. A has coordinate (2, –1) and A’ has coordinates A’ A Example 4: Art History Application The figure shows part of a tile floor. Write a rule for the translation of hexagon 1 to hexagon 2.

  14. Step 2 Translate. To translate A to A’, 2 units are subtracted from the x-coordinate and 1 units are added to the y-coordinate. Therefore, the translation rule is (x, y) → (x – 3, y + 1 ). A’ A Example 4 Continued The figure shows part of a tile floor. Write a rule for the translation of hexagon 1 to hexagon 2.

  15. A’ Check It Out! Example 4 Use the diagram to write a rule for the translation of square 1 to square 3. Step 1 Choose two points. Choose a Point A on the preimage and a corresponding Point A’ on the image. A has coordinate (3, 1) and A’ has coordinates (–1, –3).

  16. A’ Check It Out! Example 4 Continued Use the diagram to write a rule for the translation of square 1 to square 3. Step 2 Translate. To translate A to A’, 4 units are subtracted from the x-coordinate and 4 units are subtracted from the y-coordinate. Therefore, the translation rule is (x, y)  (x – 4, y – 4).

  17. Reflections 9-1

  18. An isometry is a transformation that does not change the shape or size of a figure. Reflections, translations, and rotations are all isometries. Isometries are also called congruence transformations or rigid motions. Recall that a reflection is a transformation that moves a figure (the preimage) by flipping it across a line. The reflected figure is called the image. A reflection is an isometry, so the image is always congruent to the preimage.

  19. Example 1: Identifying Reflections Tell whether each transformation appears to be a reflection. Explain. B. A. No; the image does not Appear to be flipped. Yes; the image appears to be flipped across a line..

  20. Check It Out! Example 1 Tell whether each transformation appears to be a reflection. a. b. Yes; the image appears to be flipped across a line. No; the figure does not appear to be flipped.

  21. Draw a segment from each vertex of the preimage to the corresponding vertex of the image. Your construction should show that the line of reflection is the perpendicular bisector of every segment connecting a point and its image.

  22. Example 2: Drawing Reflections Copy the triangle and the line of reflection. Draw the reflection of the triangle across the line. Step 1 Through each vertex draw a line perpendicular to the line of reflection.

  23. Example 2 Continued Step 2 Measure the distance from each vertex to the line of reflection. Locate the image of each vertex on the opposite side of the line of reflection and the same distance from it.

  24. Example 2 Continued Step 3 Connect the images of the vertices.

  25. Check It Out! Example 2 Copy the quadrilateral and the line of reflection. Draw the reflection of the quadrilateral across the line.

  26. 1 Understand the Problem Example 3: Problem-Solving Application Two buildings located at A and B are to be connected to the same point on the water line. Where should they connect so that the least amount of pipe will be used? The problem asks you to locate point X on the water line so that AX + XB has the least value possible.

  27. Make a Plan Let B’ be the reflection of point B across the water line. For any point X on the water line, so AX + XB = AX + XB’. AX + XB’ is least when A, X,and B’are collinear. 2 Example 3 Continued

  28. 3 Solve Reflect B across the water line to locate B’. Draw and locate X at the intersection of and the water line. Example 3 Continued

  29. 4 Look Back Example 3 Continued To verify your answer, choose several possible locations for X and measure the total length of pipe for each location.

  30. and would be congruent. Check It Out! Example 3 What if…? If A and B were the same distance from the river, what would be true about and ? A B River X

  31. X(2,–1) X’(2, 1) Y(–4,–3) Y’(–4, 3) Z(3, 2) Z’(3, –2) Example 4A: Drawing Reflections in the Coordinate Plane Reflect the figure with the given vertices across the given line. X(2, –1), Y(–4, –3), Z(3, 2); x-axis The reflection of (x, y) is (x,–y). Y’ Z X’ X Z’ Y Graph the image and preimage.

  32. S’ T’ R S R(–2, 2) R’(2, –2) T R’ S(5, 0) S’(0, 5) T(3, –1) T’(–1, 3) Example 4B: Drawing Reflections in the Coordinate Plane Reflect the figure with the given vertices across the given line. R(–2, 2), S(5, 0), T(3, –1); y = x The reflection of (x, y) is (y, x). Graph the image and preimage.

  33. S V U T S(3, 4) S’(3, –4) T’ U’ T(3, 1) T’(3, –1) U(–2, 1) U’(–2, –1) S’ V’ V(–2, 4) V’(–2, –4) Check It Out! Example 4 Reflect the rectangle with vertices S(3, 4), T(3, 1), U(–2, 1) and V(–2, 4) across the x-axis. The reflection of (x, y) is (x,–y). Graph the image and preimage.

  34. Lesson Quiz: Part I 1. Tell whether the transformation appears to be a reflection. yes 2. Copy the figure and the line of reflection. Draw the reflection of the figure across the line.

  35. Lesson Quiz: Part II Reflect the figure with the given vertices across the given line. 3.A(2, 3), B(–1, 5), C(4,–1); y = x A’(3, 2), B’(5,–1), C’(–1, 4) 4.U(–8, 2), V(–3, –1), W(3, 3); y-axis U’(8, 2), V’(3, –1), W’(–3, 3) 5.E(–3, –2), F(6, –4), G(–2, 1); x-axis E’(–3, 2), F’(6, 4), G’(–2, –1)

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