OVERVIEW Properties & Characteristics of Dilation G.SRT.1

Transcription

1 VERVIEW Properties & haracteristics of Dilation G.SRT.1 G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor: a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Similarity is directly connected to the dilation transformation and so it is essential to build a strong understanding of the properties and characteristics of dilation. Students need to look at the effects of dilation on lines and point as well as different scale factors and centers. Verify experimentally the properties of dilations given by a center and a scale factor: the dilation of a line segment is longer or shorter depending on the ratio given by the scale factor. Dilations are the foundation of the entire unit. Take the time to really develop the properties. (1) The student will determine the properties of dilation. (2) The student will be able to dilate when the center of dilation is in, on or out of the shape. (3) The student will be able dilate when given a center of dilation and a scale factor. (4) The student will be able to determine the center of dilation and the scale factor from a diagram. (5) The student will be able to use the dilation coordinate rules for dilations using any center of dilation. The purpose of learning dilations is to prepare the students for similarity. Students need to feel comfortable with dilations that create enlargements and reductions. Dilations centered at a vertex create the classic overlapping triangle diagram that shows up regularly in similarity. The definition of dilation also helps establish relationships about angles and parallel lines later. DN T SHRT UT THIS RE. Spend time developing a strong basis in dilation skill and understanding. 1 Focus on the definition of dilation. It will help establish relationships in similarity. 2 Enlargements & Reductions. Students struggle with getting ratios correct and this focus will help students learn the proper way to write a scale factor. 3 Work ackwards. To test understanding start with a dilation and have students determine its scale factor and center. If students can work backwards, then they understand the process. gsrt1properties haracteristicsofdilations Page 1 of 8 4/29/2014

2 NTES Properties & haracteristics of Dilation G.SRT.1 NEPT 1 Dilation Properties So what actually happens when a shape is dilated? The length of each side of the image is equal to the length of the corresponding side of the pre-image multiplied by the scale factor, k; = k, = k and = k. This dilation has a scale factor of 1:3. : 4: 12 1:3 4 cm 5 cm 5.5 cm ' 12 cm 15 cm ' 16.5 cm : 5.5: :3 : 5:15 1:3 DEFINITIN dilation with center and a scale factor of k is a transformation that maps every point P in the plane to point P so that the following properties are true. 1. If P is NT the center, then the P lies on P. The scale factor k is a positive number such that k = and k 1. Enlargement P P' P ' P P ' P > 1 2. If P is the center point, then P = P. The center of dilation is the only point in the plane that does not move. = P = P' Reduction P' P 0 < P ' P < 1 PRPERTIES DILTIN PRPERTIES - dilation is NT an isometric transformation so its properties differ from the ones we saw with reflection, rotation and translation. The following properties are preserved between the pre-image and its image when dilating: ngle measure (angles stay the same) Parallelism (things that were parallel are still parallel) ollinearity (points on a line, remain on the line) Distance IS NT preserved!!! fter a dilation, the pre-image and image have the same shape but not the same size. gsrt1properties haracteristicsofdilations Page 2 of 8 4/29/2014

3 NTES Properties & haracteristics of Dilation G.SRT.1 NEPT 2 - Verify experimentally the properties of dilations given by a center and a scale factor: a dilation takes a line passing through the center of dilation unchanged. Dilation of when the center of dilation is point on the line with scale factor of k, k > 1. PRE-IMGE ( ) IMGE ( = ) ' ' ' original Longer dilation Dilation of when the center of dilation is point on the line with scale factor of k, 0 < k < 1. PRE-IMGE ( ) IMGE ( = ) ' ' original ' ' Shorter dilation ' Dk, ( ) ( ) The result of dilating by a scale factor of k > 1 with the center of dilation that is on the is simply. The dilation maps all points either on to or which are the opposite rays that form when is between and. Dk, ( ) ( ) The result of dilating by a scale factor of 0 < k < 1 with the center of dilation that is on the is simply. The dilation maps all points either on to or which are the opposite rays that form when is between and. gsrt1properties haracteristicsofdilations Page 3 of 8 4/29/2014

4 NTES Properties & haracteristics of Dilation G.SRT.1 NEPT 3 - Verify experimentally the properties of dilations given by a center and a scale factor: a dilation takes a line not passing through the center of the dilation to a parallel line. PRE-IMGE ( ) IMGE ( ) ' ' ' ' The center of dilation is point (not on ) and the scale factor is k > 1. ngles are preserved in dilation, thus and because these corresponding angles are congruent, ' '. It is critical to understand that dilations create parallel lines between LL pre-image and image corresponding segments and lines. gsrt1properties haracteristicsofdilations Page 4 of 8 4/29/2014

5 NTES Properties & haracteristics of Dilation G.SRT.1 NEPT 4 The coordinate rule for dilation. There is not much to discover here because the coordinate rule is provided in the definition, Dk, ( x, y) ( kx, ky) When is the origin This was already discovered intuitively in G..2 when investigating isometric and non-isometric transformations. It was discovered that if one or two variables was multiplied by a value other than 1 or -1 there was no longer an isometry. The difference between the dilation and the stretch was also established. Dilations must have the same value multiplied to both variables, whereas the stretch has different values.,5 DILTIN D ( x, y) (5 x,5 y) D ( x, y) ( x, y) , 3 D (, ) ( 3, 3 ), 3 x y x y STRETH Using S as the notation for stretch: S( x, y) (2 x,3 y) Indicates a two-way stretch. Double horizontal and triple vertical. 2 S( x, y) ( x,5 y) 3 S( x, y) (8 x, y) Indicates a one-way horizontal stretch. It is obvious that for a dilation to maintain its proportionality of sides, the two variables must be multiplied by a constant value, k, known as the scale factor. gsrt1properties haracteristicsofdilations Page 5 of 8 4/29/2014

6 SSESSMENT Properties & haracteristics of Dilation G.SRT.1 1. Which of the following is a dilation? ) T( x, y) ( x 4, y 3) ) T( x, y) ( y, x) ) T( x, y) (2 x,2 y) D) T( x, y) (5 x,3 y) 2. Which of the following is a non-isometric transformation? ) Rotation ) Dilation ) Reflection D) Translation 3. Which is not an example of dilation? ) The pupil of your eye reducing in size due to the bright sun. ) Using a graphics program to alter a pictures size from 8 by 10 to 4 by 5. ) Standing in front of a circus fun house mirror that makes you very tall. D) Using a shrink gun to make you one-fourth the size that you are now. 4. If we D,2 then the correct diagram would be: riginal ) ) ) D) m m m' m m' m m' m 5. Determine the scale factor that best suits the provided diagram ( is the center of dilation). ' ' ) 2 ) 1 2 ) 1 3 D) Determine the scale factor of the given dilation from point? ) 1 : 2 ) 2 : 1 ) 2 : 5 D) 5 : 2 ' 2.5 cm 5 cm gsrt1properties haracteristicsofdilations Page 6 of 8 4/29/2014

7 SSESSMENT Properties & haracteristics of Dilation G.SRT.1 7. Given D,2 P( x, y) P'( 4,10) then P(x, y) is ) P(-8, 20) ) P(-2,5) ) P(-4, 12) D) P(0, 20) 8. Determine whether the dilation is an enlargement or a reduction and then determine the scale factor. Place the scale factor in the most reduced form (no decimals). a) Enlarge or Reduce b) Enlarge or Reduce c) Enlarge or Reduce d) Enlarge or Reduce I' I ' 2 cm 6 cm 4 cm 6 cm 7 cm D ' 4 cm D' 2 cm ' 1 cm e) Enlarge or Reduce f) Enlarge or Reduce g) Enlarge or Reduce h) Enlarge or Reduce ' 4.5 cm 13.5 cm P 1 cm P' 7 cm 3 cm 6 cm ' I' I 2 cm ' 1 cm 9. Dilate the following. ( is the origin). D (2, 1) (, ) b) D, 3 ( 2,4) (, ) c) D 1 (3,12) (, ),3 a), D (, ) (, ) e) D 3 (8, 4) (, ) f) D,2.5(10,4) (, ) 4 8, d),4 4 gsrt1properties haracteristicsofdilations Page 7 of 8 4/29/2014

8 SSESSMENT Properties & haracteristics of Dilation G.SRT.1 nswers: 1) 2) 3) 4) D 5) 6) 7) 8) a) Reduce, 3 : 1 b) Reduce, 4 : 1 c) Reduce, 3 : 2 d) Enlarge, 2 : 3 e) Reduce, 3 : 1 f) Enlarge, 1 : 3 g) Reduce, 3 : 2 h) Reduce, 3 : 2 9) a) (4, -2) b) (6, -12) c) (1, 4) d) (5, 6) e) (-6, 3) f) (25, 10) gsrt1properties haracteristicsofdilations Page 8 of 8 4/29/2014