: Properties of Dilations and Equations of lines Opening Exercise Identify which dilation figures were created using r = 1, using r > 1, and using 0 < r < 1.
: Properties of Dilations and Equations of lines Learning Targets I can identify the properties of dilation mentioned as followed: 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. I can use the dilation theorem to show that the scale drawings constructed using the ratio have a scale factor that is the same as the scale factor for the dilation. Example 1. Let line L be a line not passing through the center of dilation. Dilate line L, with a scale factor r = 2 from center O. Step 1. To dilate the line, we will choose two points on L (points P and Q) to dilate. Step 2. Draw rays from center O through each of the points P and Q Step 3. Use our compass to measure the distance from O to P Step 4. Mark along the ray OP (without changing the radius of the compass) to mark P. Recall that the dilated point P is the distance 2 OP Repeat the process for Q What is the relationship between segment PQ and P Q? Explain your answer Since the lengths were measured proportionate then PQ is a side splitter and PQ is P Q What would happen if the line goes through the center of dilation? Example 2. Let line L be a line passing through the center of dilation. Dilate line L, with a scale factor r = 2 from center O. Conclusion: 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.
Example 3 a. Dilate using constructions AB about the center O(0,0) and scale factor of 2 8 6 4 A (1,4) b. What are the coordinates of points A and B 2 B (3,1) O (0,0) 5 c. Describe a rule for dilating coordinates about the origin c. Write the equation of AB in y = mx + b (use Stats on calculator) d. Write the equation of A B (use Stats on calculator) e. What can we say about the slopes of AB and A B f. What can you say about the y-intercepts? Explain
: Properties of Dilations and Equations of lines Classwork Suppose a dilation is centered at the origin. You can find the dilation image of a point by multiplying its coordinates by the scale factor. Notation D 2 (A) = A Scale factor 2, (x, y) (2x, 2y) 1. The triangle ABC has coordinates A = (6, 1), B = (12, 4), and C = ( 6, 2). The triangle is dilated from the origin by a scale factor r = 1. Identify the coordinates of the dilated triangle A B C. 2 2. Figure DEFG has coordinates D = (1, 1), E = (7, 3), F = (5, 4), and G = ( 1, 4). The figure is dilated from the origin by scale factor r = 7. Identify the coordinates of the dilated figure D E F G. 3. A dilation has center (0, 0). Find the image of each point for the given scale factor. 1. A(3, 4); D 7 (A) 2. B(0, 4); D 3.4 (B) 3. C(5, 6); D 5 3 (C) 4. Graph quadrilateral ABCD and its image A B C D for a dilation with center (0, 0) and a scale factor of The vertices of ABCD are: A(2, 3), B( 2, 4), C( 3, 2), D(3, 3).
5. Graph pentagon JKLMN and its image J K L M N for a dilation with center (0,0) and a scale factor of 1.5. The vertices of JKLMN are: J(0, 3), K(3, 3), L(3, 0), M(0, 3), N( 1, 0). 6. In the coordinate plane, line m has a slope of 2 and a y-intercept of (0, 5). Line n is the result of dilating line m by a scale factor of 4 with a center of (0,0). What are the slope and y-intercept of line n? a. Line n has a slope of ½ and a y-intercept of (0, 3). b. Line n has a slope of 2 and a y-intercept of (0, 5). c. Line n has a slope of 2 and a y-intercept of (0, 20). d. Line n has a slope of 8 and a y-intercept of (0, 20). 7. Line segment CD with endpoints C( 5,16) and D( 20, 4) lies in the coordinate plane. The segment will be dilated with a scale factor of 2 and a center at the origin to create C'D'. What will equation of C'D'? 8. Line segment CD with endpoints C( 4,16) and D( 20,4) lies in the coordinate plane. The segment will be dilated with a scale factor of ¼ and a center at the origin to create C'D'. What equation of the line C'D'? 9. In the coordinate plane, line m has a slope of ¼ and a y-intercept of (0, -3). Line n is the result of dilating line m by a scale factor of 2 with a center of (0, 0). What are the slope and y-intercept of line n? a. Line n has a slope of ¼ and a y-intercept of (0, -3). b. Line n has a slope of ¼ and a y-intercept of (0, -6). c. Line n has a slope of ¾ and a y-intercept of (0, -2). d. Line n has a slope of 2.25 and a y-intercept of (0, -6).