• Students learn the definition of dilation and why “same shape” is not good enough to say two figures are similar.
• Students know that dilations magnify and shrink figures.
Definition: A dilation is a transformation of the plane with center 0, while scale factor (r > 0) is a rule that
assigns to each point P of the plane a point Dilation(P) so that:
1. Dilation(0) = 0, (i.e., a dilation does not move the center of dilation.)
2. If P ≠ 0, then the point dilation(P), (to be denoted more simply by P") ) is the point on the ray OP so that |OP'| = r|OP|.
In other words, a dilation is a rule that moves points in the plane a specific distance, determined by the scale factor r, from a center 0. When the scale factor r > 1, the dilation magnifies a figure. When the scale factor 0 < r < 1, the dilation shrinks a figure. When the scale factor r = 1, there is no change in the size of the figure, that is, the figure and its image are congruent.
Two geometric figures are said to be similar if they have the same shape but not necessarily the same size. Using that informal definition, are the following pairs of figures similar to one another? Explain.
1. Given |OP| = 5 in.
a. If segment OP is dilated by a scale factor r = 4, what is the length of segment OP'?
b. If segment OP is dilated by a scale factor r = 1/2, what is the length of segment OP'?
Use the diagram below to answer Exercises 2–6. Let there be a dilation from center 0. Then, dilation(P) = P' and dilation(Q) = Q'. In the diagram below, |OP| = 3 cm and |OQ| = 4 cm as shown.
2. If the scale factor is r = 3, what is the length of segment OP'?
3. Use the definition of dilation to show that your answer to Exercise 2 is correct.
4. If the scale factor is r = 3, what is the length of segment OQ'?
5. Use the definition of dilation to show that your answer to Exercise 4 is correct.
6. If you know that |OP| = 3, |OP'| = 9, how could you use that information to determine the scale factor?
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