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# 6.7 similarity transformations and coordinate geometry

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### 6.7 similarity transformations and coordinate geometry

1. 1. 6.7 Similarity Transformations and Coordinate Geometry 6.7 Bell Thinger Give the coordinates of a point twice as far from the origin along a ray from (0, 0). 1. (3, 5) ANSWER (6, 10) 2. (–2, 0) ANSWER (–4, 0) Give the coordinates of a point one-half as far from the origin along a ray from (0, 0). 3. (6, –4) ANSWER (3, –2) 4. (0, –7) ANSWER (0, –3.5)
2. 2. 6.7
3. 3. Example 1 6.7 Draw a dilation of quadrilateral ABCD with vertices A(2, 1), B(4, 1), C(4, – 1), and D(1, – 1). Use a scale factor of 2. SOLUTION First draw ABCD. Find the dilation of each vertex by multiplying its coordinates by 2. Then draw the dilation. (x, y) (2x, 2y) A(2, 1) L(4, 2) B(4, 1) M(8, 2) C(4, –1) N(8, –2) D(1, –1) P(2, –2)
4. 4. Example 2 6.7 A triangle has the vertices A(4,– 4), B(8, 2), and C(8,– 4). The image of ABC after a dilation with a scale factor of 1 is DEF. 2 ABC and DEF. a. Sketch SOLUTION a. The scale factor is less than one, so the dilation is a reduction. 1 x, 1 y (x, y) 2 2 A(4, – 4) D(2, – 2) B(8, 2) E(4, 1) C(8, – 4) F(4, – 2)
5. 5. Example 2 6.7 A triangle has the vertices A(4,– 4), B(8, 2), and C(8,– 4). The image of ABC after a dilation with a scale factor of 1 is DEF. 2 b. Verify that ABC and DEF are similar. SOLUTION b. Because ∠C and ∠F are both right angles, ∠C ≅ ∠F. Show that the lengths of the sides that include ∠C and ∠F are proportional. Find the horizontal and vertical lengths from the coordinate plane. AC ? BC 4 6 2 = 3 DF = EF So, the lengths of the sides that include ∠C and ∠F are proportional.
6. 6. Guided Practice 6.7 Find the coordinates of L, M, and N so that LMN is a dilation of PQR with a scale factor of k. Sketch PQR and LMN. 1. P(–2, -1), Q(–1, 0), R(0, –1); k = 4 ANSWER L (–8, –4), M (– 4, 0), N (0, –4)
7. 7. Guided Practice 6.7 Find the coordinates of L, M, and N so that LMN is a dilation of PQR with a scale factor of k. Sketch PQR and LMN. 2. P(5, –5), Q(10, –5), R(10, 5); k = 0.4 ANSWER L (2, –2), M (4, –2), N (4, 2)
8. 8. Example 3 6.7 Photo Stickers You are making your own photo stickers. Your photo is 4 inches by 4 inches. The image on the stickers is 1.1 inches by 1.1 inches. What is the scale factor of the reduction? SOLUTION The scale factor is the ratio of a side length of the sticker image to a side length of the original photo, or 11 1.1 in. . In simplest form, the scale factor is . 40 4 in.
9. 9. Example 4 6.7 SOLUTION Determine if EFGH is a dilation of PQRS by checking whether the same scale factor can be used to obtain E, F, and G from P, Q, and R.
10. 10. Example 4 6.7 (x, y) (kx, ky) P(3, 0) E(9, 0) k=3 Q(1, 1) F(3, 3) k=3 R(0, 2) G(0, 6) k=3 Because k is the same in each case, the image is a dilation with a scale factor of 3. So, you can use the scale factor to find the image H of point S. S(4, 5) H(3 . 4, 3 . 5) = H(12, 15) ANSWER The correct answer is C. CHECK: Draw rays from the origin through each point and its image.
11. 11. Guided Practice 6.7 3. Suppose a figure containing the origin is dilated. Explain why the corresponding point in the image of the figure is also the origin. ANSWER A dilation with respect to the origin and scale factor k can be described as (x, y) (kx ,ky). If (x, y) = (0, 0) then (kx, ky) = (k . 0, k . 0) = (0, 0).
12. 12. Exit Slip 6.7 1. Draw a dilation with scale factor 2 of ABC with vertices A(1, 0), B(0, 3) and C(3, 2). Label the image DEF. ANSWER
13. 13. Exit Slip 6.7 2. ABC has vertices A(0, 0), B(0, 8) and C(6, 0). Draw a dilation of ABC using a scale factor of 1 . Label 2 the image DEF. ANSWER
14. 14. Exit Slip 6.7 PS 3. In the diagram, MR = . find MN and MR. SN RN ANSWER 48; 30
15. 15. Exit Slip 6.7 ANSWER Yes; XYZ ~ CBA ANSWER No
16. 16. 6.7 Homework Pg 428-431 #6, 11, 16, 19, 25