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Understanding Geometry Transformations: Translation, Rotation, Reflection, Dilation

Learn about different transformations in geometry - translation, rotation, reflection, dilation. Master the techniques with examples and interactive exercises. Understand the concepts clearly to ace geometry tests!

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Understanding Geometry Transformations: Translation, Rotation, Reflection, Dilation

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  1. TRANSFORMATIONS!

  2. To transform something is to change it. In geometry, there are specific ways to describe how a figure is changed. • Translation • Rotation • Reflection • Dilation

  3. Renaming Transformations Capitol letters To name transformed shapes we use the same letters with a _____ symbol: We use __________ _______ to help name the original shapes. Prime Before: PRE IMAGE After: IMAGE

  4. Atranslation"slides" an object a fixed distance in a given direction.  Translations are ISOMETRIC – which means they have the same size and the same shape! Translations are SLIDES.

  5. Let's examine some translations related to coordinate geometry. The example shows how each vertex moves the same distance in the same direction.

  6. A rotation is a transformation that turns a figure about a fixed point called the ___________________.  A Rotation is also ISOMETRIC Center of rotation

  7. Clockwise Counterclockwise This rotation is 90 degrees counterclockwise.

  8. Areflectioncan be seen in water, in a mirror, in glass, or in a shiny surface. A Reflection is ISOMETRIC

  9. The line (where a mirror may be placed) is called the line of reflection.  The distance from a point to the line of reflection is the same as the distance from the point's image to the line of reflection. A reflection can be thought of as a "flipping" of an object over the line of reflection. If you folded the two shapes together line of reflection the two shapes would overlap exactly!

  10. Adilationis NOT ISOMETRIC! Why? A dilation used to create an image larger than the original is called an enlargement.  A dilation used to create an image smaller than the original is called a reduction.

  11. TRANSLATIONS!

  12. length direction A vector has _______ and _________ – it tells you where to go! The vector <7,4> tells your x value to change by +7 and your y value to change by + 4

  13. Using points P,E,N,T,A, translate them along the vector –5, –1. P(1, 0), E(2, 2), N(4, 1), T(4, –1), and A(2, –2) The vector indicates a translation 5 units left and 1 unit down. (x, y)→ (x – 5, y – 1) P(1, 0)→ (–4, –1) E(2, 2)→ (–3, 1) N(4, 1)→ (–1, 0) T(4, –1)→ (–1, –2) A(2, –2)→ (–3, –3)

  14. EX 2: Translate GHJK with the vertices G(–4, –2), H(–4, 3), J(1, 3), K(1, –2) along the vector 2, –2. Choose the correct coordinates for G'H'J'K'. A.G'(–6, –4), H'(–6, 1), J'(1, 1), K'(1, –4) B.G'(–2, –4), H'(–2, 1), J'(3, 1), K'(3, –4) C.G'(–2, 0), H'(–2, 5), J'(3, 5), K'(3, 0) D.G'(–8, 4), H'(–8, –6), J'(2, –6), K'(2, 4)

  15. The graph shows repeated translations that result in the animation of the raindrop. Describe the translation of the raindrop from position 3 to position 4 using a translation vector < a , b >. (–1 + a, –1 + b) or (–1, –4) –1 + a = –1 –1 + b = –4 a = 0 b = –3 Answer: translation vector:

  16. REFLECTIONS

  17. What happens to points in a Reflection? • Name the points of the PRE-IMAGE: A ( , ) B ( , ) C ( , ) • Name the points of the IMAGE: A’ ( , ) B’ ( , ) C’ ( , ) • What is the line of reflection? • How did the points change from the original to the reflection?

  18. equidistant A reflection is always ____________from the ___________________! Line of reflection

  19. EX 1: Reflect ABCD with vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3) across the x-axis: Multiply the y-coordinate of each vertex by –1. (x, y)→ (x, –y) A(1, 1)→ A'(1, –1)B(3, 2)→ B'(3, –2)C(4, –1)→ C'(4, 1)D(2, –3)→ D'(2, 3)

  20. EX 2: Graph quadrilateral ABCD with vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3) and its reflected image in the y-axis. Multiply the x-coordinate of each vertex by –1. (x, y)→ (–x, y) A(1, 1)→ A'(–1, 1)B(3, 2)→ B'(–3, 2)C(4, –1)→ C'(–4, –1)D(2, –3)→ D'(–2, –3)

  21. Ex 3: Given the Pre Image Quadrilateral ABCD with vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3). Graph ABCDand its image under reflection of the line y = x. Interchange the x- and y-coordinates of each vertex. (x, y)→ (y, x) A(1, 1)→ A'(1, 1)B(3, 2)→ B'(2, 3)C(4, –1)→ C'(–1, 4)D(2, –3)→ D'(–3, 2)

  22. EX 4: Quadrilateral EFGH has vertices E(–3, 1), F(–1, 3), G(1, 2), and H(–3, –1). The quadrilateral was reflected, giving the image the coordinates: How was the quadrilateral reflected? E'(1, –3), F'(3, –1), G'(2, 1), H'(–1, –3) It was reflected over the line y=x, since all of the x and y values have been switched

  23. DILATIONS

  24. Dilations always involve a change in size. How did the coordiates of the image compare to the coordinates of the pre image?

  25. If k>1 the dilation is an _________________ If k<1 the dilation is a __________________ Does it make sense for k to be less than 0? Enlargement Reduction

  26. EX 1: Trapezoid EFGH has vertices E(–8, 4), F(–4, 8), G(8, 4) and H(–4, –8). Graph the image of EFGH after a dilation centered at the origin with a scale factor (k) of Multiply the x- and y-coordinates of each vertex by the scale factor,

  27. Ex 2: Triangle ABC has vertices A(–2, -4), B(3, 2), C(0, 4). Graph the image of ABC after a dilation centered at the origin with a scale factor of 1.5.

  28. Rotations and Symmetry

  29. Rotation – a transformation that turns every point of a pre-image through a specified angle and direction about a fixed point, called the Center of Rotation. P is the Center of rotation!

  30. Angle of rotation – the angle between a pre-image point and its corresponding image point.. The Angle of Rotation from A to A’ is 90° Counterclockwise!

  31. EX 1: For the diagram to the right, which description best identifies the rotation of triangle ABC around point Q? A. 20° clockwise B. 20° counterclockwise C. 90° clockwise D. 90° counterclockwise

  32. Thankfully, we really only care about rotating three ways…

  33. EX 2: Hexagon DGJTSR is shown below. What is the image of point T after a 90 counterclockwise rotation about the origin? A (5, –3) B (–5, –3) C (–3, 5) D (3, –5)

  34. EX 3: • What is the image of point Q after a 90° counterclockwise rotation about the origin? • What is the image of point Q after a 180° counterclockwise rotation about the origin? • What is the image of point Q after a 270° counterclockwise rotation about the origin? (–5, 4) (-4, -5) (5, -4)

  35. Rotational Symmetry: A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180⁰ or less. The figure above has rotational symmetry because it maps onto itself by a rotation of 90⁰.

  36. Whena figure can be rotated less than 360° and the image and pre-image are indistinguishable (the same) Symmetry Rotational: 120° 90° 60° 45° Lines of : 3 4 6 8 Symmetry? 360/3 = 120 360/4 = 90 360/6 = 60 360/8 = 45

  37. Does the following shape have symmetry? Lines of Symmetry? _______________ Rotational Symmetry? _______________

  38. Day 5 Composites!

  39. Composite transformations mean: More than one transformation is happening! Original Move 1 Move 2

  40. EX 1: Quadrilateral BGTS has vertices B (–3, 4), G (–1, 3), T(–1 , 1), and S (–4, 2). Graph BGTS and its image after a translation along 5, 0 and a reflection in the x-axis.

  41. Step 1 translation along 5, 0 (x, y) →(x + 5, y) B(–3, 4)→ B'(2, 4) G(–1, 3)→ G'(4, 3) S(–4, 2)→ S'(1, 2) T(–1, 1)→ T'(4, 1)

  42. Step 2 reflection in the x-axis (x, y)→(x, –y) B'(2, 4)→ B''(2, –4) G'(4, 3)→ G''(4, –3) S'(1, 2)→ S''(1, –2) T'(4, 1)→ T''(4, –1)

  43. Ex 2: Quadrilateral RSTU has vertices R(1, –1), S 4, –2), T (3, –4), and U (1, –3). Graph RSTU and its image after a translation along –4, 1and a reflection in the x-axis. Which point is located at (–3, 0)? A.R' B.S' C.T' D.U'

  44. EX 3: ΔTUV has vertices T(2, –1), U(5, –2), and V(3, –4). Graph ΔTUV and its image after a translation along –1 , 5 and a rotation 180° about the origin.

  45. Step 1 translation along –1 , 5 (x, y) →(x + (–1), y + 5) T(2, –1)→ T'(1, 4) U(5, –2)→ U'(4, 3) V(3, –4)→ V'(2, 1)

  46. Step 2 rotation 180 about the origin (x, y)→(–x, –y) T'(1, 4)→ T''(–1, –4) U'(4, 3)→ U''(–4, –3) V'(2, 1)→ V''(–2, –1)

  47. EX 4: ΔJKL has vertices J(2, 3), K(5, 2), and L(3, 0). Graph ΔTUV and its image after a translation along 3, 1and a rotation 180° about the origin. What are the new coordinates of L''? A. (–3, –1) B. (–6, –1) C. (1, 6) D. (–1, –6)

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