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Geom 7point4and5

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Geom 7point4and5

  1. 1. Translations & Vectors, Glide Reflections & Compositions <ul><li>Objectives: </li></ul><ul><li>Identify and use translations </li></ul><ul><li>Identify glide reflections in a plane. </li></ul><ul><li>Represent transformations as compositions of simpler transformations. </li></ul>
  2. 2. Translations <ul><li>If I move a figure over without changing the size, angles, or orientation, that is a translation . </li></ul>
  3. 3. Translations <ul><li>A translation is a transformation that maps every two points P and Q in the plane to points P’ and Q’ so that: </li></ul><ul><ul><li>PP’ = QQ’ </li></ul></ul><ul><ul><li>PP’ || QQ’, or PP’ and QQ’ are collinear </li></ul></ul>P P’ Q Q’
  4. 4. Translation Theorem <ul><li>A translation is an isometry. </li></ul>
  5. 5. Another Theorem <ul><li>Look at the picture at the bottom of p. 421 </li></ul><ul><li>2 reflections = a translation </li></ul><ul><li>If lines k and m are parallel, then a reflection in line k followed by a reflection in m is a translation. </li></ul><ul><li>If P” is the image of P, then the following is true: </li></ul><ul><li>Line PP” is perpendicular to k and m. </li></ul><ul><li>PP” = 2d where d is the distance between k & m. </li></ul>
  6. 6. Using this theorem <ul><li>In your book, look at the picture on the top of p. 422. </li></ul><ul><li>What segments are congruent? </li></ul><ul><li>Does AC = BD? </li></ul><ul><li>How long is GG” </li></ul>
  7. 7. Using this theorem <ul><li>Look at the picture in the middle of p. 422. </li></ul><ul><li>What is the horizontal shift? </li></ul><ul><li>What is the vertical shift? </li></ul><ul><li>Translations can be described as </li></ul><ul><li>(x, y) --> (x+a, y+b) </li></ul><ul><li>Where each point shifts a units horizontally and b units vertically </li></ul>
  8. 8. Using this theorem <ul><li>Look at the picture at the bottom of p. 422. </li></ul><ul><li>What is the horizontal shift? </li></ul><ul><li>-3 </li></ul><ul><li>What is the vertical shift? </li></ul><ul><li>4 </li></ul>
  9. 9. Translations Using Vectors <ul><li>A vector is a quantity that has both direction and magnitude (size) and is represented by an arrow drawn between 2 points. </li></ul>3 5
  10. 10. Translations Using Vectors <ul><li>The initial point of this vector is P and the terminal point is Q. </li></ul><ul><li>The component form of a vector combines the vertical and horizontal components. The component form of PQ is <5, 3> </li></ul>3 5 P Q
  11. 11. Identifying Vector Components <ul><li>On p. 423, in the middle, what is the name of the vector in a.? </li></ul><ul><li>JK </li></ul><ul><li>What is the component form? </li></ul><ul><li><3,4> </li></ul><ul><li>On p. 423, in the middle, what is the name of the vector in b.? What is the component form? </li></ul><ul><li>MN <0,4> </li></ul><ul><li>On p. 423, in the middle, what is the name of the vector in c.? What is the component form? </li></ul><ul><li>TS <3,-3> </li></ul>
  12. 12. Look at p. 423 Example 4 <ul><li>Translate ∆ABC using a vector of <4,2> </li></ul><ul><li>The green arrows show each point of the triangle going over 4 and up 2. </li></ul><ul><li>The ends of these arrows create the new triangle </li></ul>
  13. 13. Look at p. 424 Example 5 <ul><li>What is the component form of the vector that can be used to describe the translation? </li></ul>
  14. 14. Classwork & Homework <ul><li>Classwork </li></ul><ul><ul><li>Page 425 16-34 evens </li></ul></ul><ul><li>Homework </li></ul><ul><ul><li>7.4 Practice B and C </li></ul></ul><ul><ul><li>When you have finished the classwork you can show me and I will give you the homework worksheets to work on </li></ul></ul>
  15. 15. Glide Reflections & Compositions <ul><li>A translation, or glide, and a reflection can be performed one after the other to produce a transformation known as a glide reflection . </li></ul><ul><li>For example, look at the bottom of p. 430. </li></ul><ul><li>The blue to the red is (x,y) --> (x+10, y) </li></ul><ul><li>The red to the green is a reflection about the x axis. </li></ul>
  16. 16. Using Compositions <ul><li>When 2 or more transformations are combined to produce a single transformation, the result is called a composition . </li></ul>
  17. 17. Composition Theorem <ul><li>The composition of 2 or more isometries is an isometry. </li></ul>
  18. 18. For example: <ul><li>If point P is at (2, -2) </li></ul><ul><li>and point Q is at (3, -4) </li></ul><ul><li>Sketch point PQ </li></ul><ul><li>Rotate PQ 90˚ counterclockwise about the origin. </li></ul><ul><li>Reflect PQ in the y axis. </li></ul><ul><li>Look at p. 431 to see this </li></ul>
  19. 19. If you switch the order . . . <ul><li>And do the reflection first and the rotation second, how does it change the result? </li></ul><ul><li>See the picture at the bottom of p. 431. </li></ul>
  20. 20. Look at p. 432 at the top <ul><li>What composition do you see? </li></ul><ul><li>1. Reflection in the line x = 2 </li></ul><ul><li>2. Rotation 90˚ clockwise about the point 2,0 </li></ul>
  21. 21. Pentominoes <ul><li>Look at the pentonimoes at the bottom of p. 432 </li></ul><ul><li>Notice how we can use rotations, reflections and translations to move 2 of the pieces on the sides to fill the gaps </li></ul><ul><li>Do p. 435 #38 </li></ul>
  22. 22. Classwork and Homework: <ul><li>Classwork: 7.5 Practice A </li></ul><ul><li>Homework: 7.5 Practice B & C </li></ul><ul><li>When you finish Practice A show me and I will give you B and C </li></ul>

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