14-1. Translations. Vocabulary. Lesson

Transcription

1 Chapter 1 Lesson 1-1 Translations Vocabular slide, translation preimage translation image congruent figures Adding fied numbers to each of the coordinates of a figure has the effect of sliding or translating the figure. BIG IDEA The Akan people of Ghana work with a tpe of cloth called adinkra on which the draw a grid with large rectangles. The then fill each rectangle with a repetitive pattern b sliding the same design along the cloth. Similar techniques can be used for other patterned tetiles. A print design can be created b selecting a few distinct square images and tiling them to create a pattern as shown below. The correspondence between an two images of the same design as in this fabric is called a slide, or translation. If ou visualize a geometric figure on a coordinate grid, a translation is determined b the change in the coordinates necessar to slide one figure onto another. 3 Some Important Geometr Ideas

2 Lesson 1-1 Horizontal or Vertical Translations Activit 1 Let C = ( 6, ), E = ( 7, 0), and D = ( 1, 3). Translate CED 7 units to the right. Step 1 Graph CED on a coordinate grid as shown below. CED is called the preimage. C = (, ) 9 preimage D = (-1, 3) E = (-7, 0) Step 2 To translate the preimage 7 units to the right, add 7 to each -coordinate. The result is a triangle 7 units to the right of CED. We call this C E D (read triangle C-prime, E-prime, D-prime ). C E D is the translation image of CED. Preimage Translate Preimage 7 Units to the Right Translation Image C = ( 6, ) ( 6 + 7, ) C = (1, ) E = ( 7, 0)?? D = ( 1, 3)?? Step 3 Graph each image point on the same grid. Each image point is 7 units to the right of the preimage point. For instance, C = (1, ) is 7 units to the right of C = ( 6, ). In general, if ou add h to each -coordinate of the points of a figure, ou will get a slide image of the original figure that is h units to the right when h is positive. If h is negative, the image will move to the left. For an preimage point (, ), the image point after the horizontal translation of h units is ( + h, ). QY QY a. To translate CED 5 units to the right, add? to the? -coordinate. b. To translate CED 6 units to the left, add? to the? -coordinate. Translations

3 Chapter 1 Activit 2 Translate C E D 9 units down. Step 1 Predict what will happen if ou add a particular number to the -coordinate. Step 2 Use our graph from Activit 1. Add 9 to each -coordinate of C E D. We call the image C E D (read triangle C-star, E-star, D-star ). C = (1, 1) is the image of C = (1, ) because (1, + 9) = (1, 1). Preimage Translate Preimage 9 Units Down Translation Image C = (1, ) (1, + 9) C =? E = (0, 0)? E =? D = (6, 3)? D =? Step 3 Graph C E D. In general, if ou add k to the second coordinate of all points in a figure, ou will slide the figure k units up when k is positive. If k is negative, as it is in Activit 2, then the figure slides down. For an preimage point (, ), the image point after a vertical translation of k units is (, + k). Congruent figures are figures with the same size and shape. A translation image is alwas congruent to its preimage. Triangles CED, C E D, and C E D are all congruent to each other. Using the smbol for is congruent to, ou can write the previous sentence as CED C E D C E D. Translations That Are Neither Horizontal nor Vertical The grid at the right shows CED and C E D. Notice that if ou add 7 to the first coordinate and add 9 to the second coordinate of each point of CED, ou get the coordinates of a point on C E D. Thus, C E D is a translation image of CED. The rule for this slide can be written as the image of (, ) is ( + 7, + 9). Because a slide image is congruent to its preimage, ou can compute the images of a few special points, like the vertices, and then use a ruler or other tools to complete the image figure. 10 C = (, ) D = (-1, 3) E = (-7, 0) -1 C* = (1, -1) D* = (6, ) -7-9 E* = (0, -9) Some Important Geometr Ideas

4 Lesson 1-1 Translation Image of An Point Under a translation h units horizontall and k units verticall, the translation image of an point (, ) is ( + h, + k). GUIDED Eample Let A = (3, ), L = (, 1), and G = (0, 3). Translate ALG 5 units up and 2 units to the left. Solution The image of (, ) under this translation is ( - 2, + 5). So: A = (3 + 2, + 5) = (1,? ). L = ( +?, 1 +? ) = (?,? ). G = (? +?,? +? ) = (?,? ) 6 G L -2 A Questions COVERING THE IDEAS In 1, complete the chart. In each case, consider the preimage point (, ). If Then the Figure Slides Coordinates of Image 1. is added to the -coordinate of ever point on the figure,?. ( +?, ) ? is added to the? -coordinate of ever point on the figure,? is added to the? -coordinate of ever point on the figure,? is added to the? -coordinate of ever point on the figure, 5. Another name for translation is?. 7.3 units up. (, +? )?. (, _ )?. ( + 5, ) 6. When ou change coordinates of points of a figure to get another figure, the original figure is called the? and the resulting figure is called its?. In 7 and, cop the grid at the right. Then graph HOG and its image under the translation that is described. 7. The image of (, ) is ( + 3, ).. The image of (, ) is ( + 3, + 2). -12 H = (,) G = (3, ) 12 O = (6,-3) Translations 6

5 Chapter 1 In 9 and 10, tell what happens to the graph of a figure when 9. k is added to the second coordinate and k is negative. 10. h is added to the first coordinate and h is positive. 11. Tell whether this statement is alwas, sometimes but not alwas, or never true: A figure and its translation image are congruent. 12. If (, ) is a preimage point, eplain what transformation ields ( + 100, + 500) as the image point. APPLYING THE MATHEMATICS 13. Polgon ABCDEFGH outlines a top view of a school building. H ( 16, 10) G ( 12, 10) D (12, 10) C (16, 10) F ( 10, ) E (10, ) A ( 16, 0) B (16, 0) The architect wishes to send this outline via to a builder. To avoid negative numbers, the architect slides the graph so that the image of point A is the origin. H G D C F E A B What are the coordinates of B, C, D, E, F, G, and H? 7 Some Important Geometr Ideas

6 Lesson a. Draw quadrilateral PQRS with P = (0, 0), Q = (6, 0), R = (6, 2), and S = (0, ). b. On the same aes, draw the image of PQRS when 2 is subtracted from each first coordinate and is subtracted from each second coordinate. c. How are the preimage and image related? 15. A B C is a slide image of ABC. A = (0, 0), B = ( 1_ 2, 0 ), C = ( 0, 1_ 3 ), and C = ( 1_, 1_ 5 ). What are the coordinates of A and B? 16. Draw three points and their images under the transformation in which the image of (, ) is ( +, - 5). 17. Suppose Q R S is the image of QRS as a result of the translation ( +.7, - 5.3). How would ou translate point (a, b) on Q R S back to QRS? a. 5; b. 6; QY ANSWER Translations