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Warm Up

Similarity Transformations and Coordinate Geometry. Warm Up. Lesson Presentation. Lesson Quiz. 3. 12. 12. (6, 10). (1, – 1). ANSWER. ANSWER. –. ,. ( – 4 , 0). ANSWER. Warm-Up. Give the coordinates of a point twice as far from the origin along a ray from (0, 0). 1. (3, 5).

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Warm Up

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  1. Similarity Transformations and Coordinate Geometry Warm Up Lesson Presentation Lesson Quiz

  2. 3. 12 12 (6, 10) (1, –1) ANSWER ANSWER – , (–4, 0) ANSWER Warm-Up Give the coordinates of a point twice as far from the origin along a ray from (0, 0). 1. (3, 5) 2. (–2, 0)

  3. (3, –2) ANSWER (0, –3.5) ANSWER Warm-Up Give the coordinates of a point one-half as far from the origin along a ray from (0, 0). 4. (6, –4) 5. (0, –7)

  4. (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) Example 1 Draw a dilation of quadrilateralABCDwith verticesA(2, 1), B(4, 1), C(4, – 1), andD(1, – 1). Use a scale factor of2. SOLUTION First draw ABCD. Find the dilation of each vertex by multiplying its coordinates by 2. Then draw the dilation.

  5. A triangle has the vertices A(4,– 4),B(8, 2), and C(8,– 4). The image of ABCafter a dilation with a scale factor of is DEF. SketchABCandDEF. 1 1 1 a. 2 2 2 a. The scale factor is less than one, so the dilation is a reduction. A(4, – 4) D(2, – 2) y x, (x, y) C(8, – 4) F(4, – 2) B(8, 2) E(4, 1) Example 2 SOLUTION

  6. b. Because Cand Fare both right angles, C F. Show that the lengths of the sides that include Cand Fare proportional. Find the horizontal and vertical lengths from the coordinate plane. A triangle has the vertices A(4,– 4),B(8, 2), and C(8,– 4). The image of ABCafter a dilation with a scale factor of is DEF. Verify that ABCand DEFare similar. ? = AC BC = 4 6 1 EF DF So, the lengths of the sides that include Cand Fare proportional. 2 3 2 b. Example 2 SOLUTION

  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. ANSWER L (–8, –4), M (– 4, 0), N (0, –4) Guided Practice 1. P(–2, 21), Q(–1, 0), R(0, –1); k = 4

  8. 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. ANSWER L (2, –2), M (4, –2), N (4, 2) Guided Practice 2. P(5, –5), Q(10, –5), R(10, 5); k = 0.4

  9. 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? The scale factor is the ratio of a side length of the sticker image to a side length of the original photo, or . In simplest form, the scale factor is . 11 40 1.1 in. 4 in. Example 3 SOLUTION

  10. Example 4 SOLUTION Determine if EFGHis a dilation of PQRSby checking whether the same scale factor can be used to obtain E, F, and G from P, Q, and R.

  11. P(3, 0) E(9, 0) Q(1, 1) F(3, 3) R(0, 2) G(0, 6) S(4, 5) H(3 4, 3 5) = H(12, 15) (x, y) (kx, ky) The correct answer is C. ANSWER Example 4 k = 3 k = 3 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. CHECK:Draw rays from the origin through each point and its image.

  12. 1 ANSWER 5 Guided Practice 3.WHAT IF? In Example 3, what is the scale factor of the reduction if your photo is 5.5 inches by 5.5 inches?

  13. ANSWER A dilation with respect to the origin and scale factor kcan be described as (x, y) (kx ,ky). If (x, y) = (0, 0)then(kx, ky) = (k 0, k 0)= (0, 0). Guided Practice 4. Suppose a figure containing the origin is dilated. Explain why the corresponding point in the image of the figure is also the origin.

  14. 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 Lesson Quiz

  15. 2. ABC has vertices A(0, 0),B(0, 8) and C(6, 0). Draw a dilation of ABC using a scale factor of . Label the image DEF. ANSWER 1 2 Lesson Quiz

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