Mathematics
Dilations
Dilations in mathematics refer to transformations that change the size of an object without altering its shape. They are performed by multiplying the coordinates of each point by a constant factor. Dilations can either enlarge or reduce the size of the original figure, and the center of dilation determines the fixed point around which the transformation occurs.
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eBook - PDF- Doug French(Author)
- 2004(Publication Date)
- Continuum(Publisher)
Chapter 7 Enlargement and Similarity ENLARGEMENT Enlargement carries the idea of making something larger, although in a mathematical sense it may also refer to making a reduction in size, a process for which there is no widely accepted word. The technical term for all such enlargements is dilatation, but it is not a word that is commonly used so I shall use the word enlargement, but it is important to note that this ambiguity about the use of the word does need explaining to students if they are not to restrict its meaning to that of everyday usage. Enlargement in this general sense is a ubiquitous feature of our world with its widespread use of maps, plans, scale drawings of all kinds, photographic enlargements, the use of photocopiers which enlarge and reduce and the corresponding facilities with text and images on a computer screen. There is no lack of examples on which to draw in helping students to appreciate the importance of the idea and its immense utility. For example, a map with a scale of 1:25,000 is an enlargement by a scale factor of two of a map with a scale of 1:50,000, while the real thing is 50,000 times larger than its depiction on the latter map. Another example is a standard sheet of A3 paper which is an enlargement by a scale factor of V2 of a sheet of A4 paper with the other paper sizes linked in a similar way. Besides the difficulty of including reductions in size, students also need to understand that enlargement not only involves a change of size, but that various properties of a shape do remain unchanged, notably the angles between lines in a shape and the proportions of the shape. The first is not difficult to appreciate, but the second is a source of considerable difficulty which is linked to an understanding of ratio and proportion, which goes beyond its application to geometry. I have discussed this general issue at length in the chapter on proportionality in this book's companion volume, Teaching and Learning Algebra (French 2002b).
eBook - PDFMathematical Practices, Mathematics for Teachers
Activities, Models, and Real-Life Examples
- Ron Larson, Robyn Silbey(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
The ratio of the dimensions of the image to the dimensions of the original figure is the scale factor of the dilation. When the scale factor is greater than 1, the dilation is an enlargement. When the scale factor is between 0 and 1, the dilation is a reduction. EXAMPLE 1 Identifying Dilations Determine whether the blue figure is a dilation of the red figure. If it is, tell whether it is a reduction or an enlargement. Then estimate the scale factor. a. b. c. SOLUTION a. Lines connecting corresponding vertices meet at a point. So, the blue figure is a dilation. Because the blue triangle is smaller, the dilation is a reduction. The dimensions of the blue triangle are about one-half the dimensions of the red triangle. So, the scale factor is about 1 — 2 . b. Lines connecting corresponding vertices meet at a point. So, the blue figure is a dilation. Because the blue square is larger, the dilation is an enlargement. The dimensions of the blue square are about twice the dimensions of the red square. So, the scale factor is about 2. c. The red figure reflects to form the blue figure. So, the blue figure is not a dilation. When an ophthalmologist examines your eyes, you may be given eye drops that cause your pupils to dilate (get larger). This allows the ophthalmologist to look way into the back of your eyes. Classroom Tip Anna Omelchenko/Shutterstock.com Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
eBook - PDF- H. S. M. Coxeter, S. L. Greitzer(Authors)
- 1967(Publication Date)
- American Mathematical Society(Publisher)
Similarities include, as special cases, isometries, for which R = 1. These remarks can be made more precise by defining a similarity to be a 6ransformation that preserves ratios of distances. For this implies that it preserves both collinearity and angles. A’B’ = kAB. Figure 4.7A The simplest kind of similarity is a diluta6ion1 which transforms each line into a parallel line. Any dilatation that is not merely a translation is called a central dilatation, because all the lines joining corresponding points of the figure and its image are concurrent. To see why this is so, examine Figures 4.7A and B, in which the corresponding segments A B and A’B’ (lying on parallel lines) satisfy the vector equation A‘B’ = &A>. Foranypoint C thatformsatrianglewith A and B, theimage C‘ is where the line through A’ parallel to AC meets the line through B’ parallel to BC. If the dilatation is not a translation, the lines A A’ and BB‘ are not parallel but meet at a point 0, such that either G’ = k z and G’ = k s , as in Figure 4.7A, or z’ = -kOA and G’ = -ROB, - - - DILATATION 95 as in Figure 4.7B. Remembering that parallel lines cut transversals into proportional segments, we easily deduce that C' lies on OC; in fact, 6? = fkZ. Varying Figure 4.7A by making 0 recede far away to the left, we see how a tratzsZatio~~ arises as the limiting form of a central dilatation A'B' = k.E when k tends to 1. Still more easily, we can change Figure 4.7B so as to make 0 the midpoint of AA'; thus the central dilatation A% = --KG includes, as a special case, the Mj-tum A'B' = -AB, d - - for which ABA'B' is a parallelogram with center 0. Figure 4.7B EXERCISES 1. What is the locus of the midpoint of a segment of varying length such that one end remains fixed while the other end runs around a circle? 2. Given an acute-angled triangle ABC, construct a square with one side lying on BC while the other two vertices lie on CA and AB, respectively.
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