CHAP. 22]
INTRODUCTION TO TRANSFORMATIONAL GEOMETRY
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Fig. 22-26 EXAMPLE 18. Finding an Unknown. Given that Dn ð8; 0Þ ¼ ð1; 0Þ, find n for a dilation in which ð0; 0Þ is the center of dilation. SOLUTION Since the origin is the center of dilation, ð1; 0Þ ¼ ðn8; n0Þ. Therefore, 8n ¼ 1 and n ¼ 13. EXAMPLE 19. Dilating a Square. Draw a square ABCD in the coordinate plane such that A ¼ ð1; 1Þ; B ¼ ð1; 2Þ; C ¼ ð2; 1Þ; and D ¼ ð2; 2Þ. Then,
(a) With O as the center of dilation, find the image of ABCD under a dilation with n ¼ 13. (b) Find the midpoint M of AB and the midpoint M 0 of A 0 B 0 : ðcÞ Find D1=3 ðMÞ: SOLUTIONS
1 1 1 (a) For a dilation with center have 1 1ð0; 2 1 Dðx, yÞ ¼0 3 x, 3 y . The image of ABCD is 0Þ0 and1 n 2¼ 3, we 0 0 0 0 0 0 A B C D , where A ¼ 3 , 3 , B ¼ ð3 , 3Þ, C ¼ 3 , 3 , and D ¼ 23 , 23 : ðbÞ M ¼ 12 ð1 þ 1Þ, 12 ð1 þ 2Þ ¼ 1, 32 and M 0 ¼ 12 ð1Þ, 12 23 ¼ 13 , 12 . ðcÞ DðMÞ ¼ M0
PROPERTIES OF TRANSFORMATIONS. We are now in a position to summarize the properties of transformations. In particular, we are interested in what is preserved under each kind of transformation. (1) (2) (3) (4)
Reflections preserve (a) distance, (b) angle measure, (c) midpoints, (d ) parallelism, and (e) collinearity. Translations preserve these same five properties, (a) through (e). Rotations preserve all five properties as well. Dilations preserve all except distance, that is, (b) through (e).
Solved Problems 22.1
Find the image of each of the following under the reflection in line t in Fig. 22-27(a): (a) point D; (b) point C, ðcÞ point B; ðd Þ AC. Ans.
ðaÞ
C
ðbÞ
D
ðcÞ
B
ðd Þ
AD