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# FOUNDATNS OF GEOM, STAT, PROB M 316L

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This 8 page Class Notes was uploaded by Reyes Glover on Sunday September 6, 2015. The Class Notes belongs to M 316L at University of Texas at Austin taught by Staff in Fall. Since its upload, it has received 59 views. For similar materials see /class/181507/m-316l-university-of-texas-at-austin in Mathematics (M) at University of Texas at Austin.

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Date Created: 09/06/15

ENLARGING FIGURES BY DILATIONS Student Handout To rotate or translate an object you can actually pick the object up and move it to its new location after the transformation To re ect the object you can at least turn it over and put it in the place it will be after the re ection 1 Will the shape of the object ever change if you translate rotate or re ect it Explain 2 Will the size of the object ever change if you translate rotate or re ect it Explain 3 Sometimes people call translations rotations and re ections rigid motions Do you think this is an appropriate name Why or why not The following exploration is to introduce you to a kind of transformation that is not a rigid motion As you work on this exploration think about these questions 0 Which features of a figure stay the same when it is enlarged 0 Which features change and how Sarah had a stylized Bevo logo that she liked and wanted to put on the front of her ring binder But it needed to be bigger and she was having a hard time drawing a larger version Her roommate Amy who was student teaching in seventh grade math said quotOh my students have just done a neat activity for enlarging a picture Why not try it Here is the method Amy39s students were using Instructions for using a quottwoband stretcherquot i Make your quottwo band stretcher by taking two rubber bands of the same size about 3quotlong and linking them together then pulling tight to make a quotknotquot ii If you are right handed tape the right handed version of the picture to be copied to your desk and tape a blank paper on its right If you are left handed use the left handed version and tape the blank paper to its left iii Use a finger of your non writing hand to hold down one end of the stretcher on the point P This point is called the center of dilation Amy39s seventh graders call it the anchor point Some people call it a projection point iv Put a crayon or pencil preferably not too sharp in the other end of the stretcher Stretch the stretcher with your crayon or pencil until the knot is on the outline you want to enlarge v Keeping the one end of the rubber band on point P draw with your crayon or pencil so that the knot always stays on the outline to be enlarged Keep your eye on the original picture not on the one you are drawing The resulting new picture is the image of the original picture under a dilation 1 Use this method to enlarge the figures on Sheet 1 and Sheet 2 Be sure to get the appropriate version of the sheet depending on whether you are right or left handed Note Don39t be dismayed if your pictures are wiggly You will develop a refinement of this procedure later 2 Compare and contrast each original figure with its image under the dilation Ignore the imperfections pretend you have drawn the images perfectly Be sure to consider all of the following o The lengths of the line segments 0 The areas of the figures 0 The angles 0 The general shape of the figure Be as detailed as possible For example don39t just say that two lengths are different tell as exactly as you can how they compare A 3 Copy triangle ABC onto a sheet of paper and choose your own center of dilation Use that center of dilation and your stretcher to enlarge the triangle Compare with other students39 choice of center of dilation Form a conjecture about what happens to the image when you change the center of dilation Test B out your conjecture and if necessary change it using a center of dilation that is higher one that is lower one closer to the triangle C and one farther from the triangle 4 Now you will figure out how to do the dilation without using the rubber bands Draw another copy of triangle ABC on a new piece of paper Pick a convenient center of dilation P not too close so the image doesn39t overlap the original and look at the rubber band as you make the image Label your image as follows Label as A39 the point you draw when the knot is on A Label B39 and C39 similarly Thinking about the rubber band can help you answer the following questions o How do the lengths PA and PA39 compare How do you know without measuring 0 How do the lengths PB and PB39 compare How do you know without measuring 0 How do the lengths PC and PC39 compare How do you know without measuring Now you have the ideas you need to figure out how to use just a ruler and pencil to form a dilation image 0 Try it first with a new copy of triangle ABC Pick your center of dilation P and use your ruler to find where points A39 B39 and C39 One you have these you can draw in the triangle A39B39C39 Try it again with a new copy of triangle ABC but this time pick the center of dilation P on the right side of the triangle if you are right handed or on the left side if you are left handed Now use this method of finding an image under a dilation to make a neater enlargement of the Bevo logo than the one you made with the rubber band stretcher Use Sheet 3 and see if you can pick your center of dilation so your image stays on the sheet 5 Now make a three band stretcher by linking three identical rubber bands together Use this stretcher to enlarge the drawings on Sheets 1 and 2 by keeping the knot closest to P on the original figure 0 How does the shape of the image compare to the shape of the original 0 How do the lengths of corresponding line segments in the original and the image compare 0 How do the areas of the image and original compare 6 There are many possible dilations Each dilation has an associated center of dilation and an associated scale factor What do you think quotscale factor means What is the scale factor for a dilation formed with a two band stretcher What is the scale factor for a dilation formed with a three band stretcher Extensions 1 Rectangle A39B39C39D39 is the image of rectangle ABCD under a certain dilation Find the center of dilation P Explain how you found it and why your method works A39 B39 2 The small rectangle A39B39C39D39 is the image of the large rectangle ABCD under a certain dilation Find the center of dilation P What can you say about the scale factor A B Note A dilation like this Where the image is smaller than the original is also called a contraction 3 Triangle ABC has the same shape as triangle ABC but can39t possibly be the image of triangle ABC under any dilation Explain Why not Hintz Try to find a center of dilation A39 A A B C 3 a Find a rotation so that if you first transform triangle ABC by that rotation then you can get triangle A39B39C39 from the transformed triangle by a dilation A39 A A B C b Find a re ection so that if you first transform triangle ABC by that re ection then you can get triangle A39B39C39 from the transformed triangle by a dilation A39 A A B C SHEET 1 Right handed Version Dilation Activity M316L SHEET 1 Left handed Version Dilation Activity M316L SHEET 2 Right handed Version Dilation Activity M316L SHEET 2 Left handed Version Dilation Activity M316L A B

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