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7-6 Similarity Transformations p. 511 You identified congruence transformations. • Identify similarity transformations. • Verify similarity after a similarity transformation. Definitions Transformation – an operation that maps an original figure (preimage) onto a new figure (image). Dilation – a transformation that enlarges or reduces the original figure proportionally. Center of dilation – a fixed point used in a dilation Scale factor of a dilation – the extent of the dilation. Scale factor – the ratio of a length on the image to a corresponding length on the preimage. The letter “k” usually represents the scale factor of a dilation. The value of “k” determines whether the dilation is an enlargement or a reduction. A dilation is a type of similarity transformation since it produces a similar figure. p. 511 Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. B is smaller than A, so the dilation is a reduction. The distance between the vertices at (2, 2) and (2, –2) for A is 4 and from the vertices at (1, 1) and (1, –1) for B is 2. Answer: So, the scale factor is _2 or _1 . 4 2 Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. B is larger than A, so the dilation is an enlargement. The distance between the vertices at (3, 3) and (–3, 3) for A is 6 and from the vertices at (1, 1) and (–1, 1) for B is 2. Answer: So, the scale factor is _6 or 3. 2 Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. A. reduction; _1 _2 _1 B. reduction; _3 C. enlargement; 2 D. enlargement; 3 B. Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. A. reduction; _2 3 _1 B. reduction; 3 _3 C. enlargement; 2 D. enlargement; 2 PHOTOCOPYING A photocopy of a receipt is 1.5 inches wide and 4 inches long. By what percent should the receipt be enlarged so that its image is 2 times the original? What will be the dimensions of the enlarged image? To enlarge the receipt 2 times the original, use a scale factor of 2. Written as a percent, the scale factor is (2 ● 100%) or 200%. Now, find the dimensions of the enlarged receipt. width: 1.5 in. ● 200% = 3 in. length: 4 in. ● 200% = 8 in. Answer: The enlarged receipt will be 3 inches by 8 inches. PHOTOGRAPHS Mariano wants to enlarge a picture he took that is 4 inches by 7.5 inches. He wants it to fit perfectly into a frame that is 400% of the original size. What will be the dimensions of the enlarged photo? A. 15 inches by 25 inches B. 8 inches by 15 inches C. 12 inches by 22.5 inches D. 16 inches by 30 inches Verifying Similarity after a Dilation If you want to verify that a dilation produces a similar figure, you can compare corresponding angles and sides. For triangles, use SAS Similarity A. Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. original: M(–6, –3), N(6, –3), O(–6, 6) image: D(–2, –1), F(2, –1), G(–2, 2) Graph each figure. Since M and D are both right angles, M D. Show that the lengths of the sides that include M and D are proportional. Use the coordinate grid to find the lengths of the vertical segments MO and DG and the horizontal segments MN and DF. Answer: Since the lengths of the sides that include M and D are proportional, ΔMNO ~ ΔDFG by SAS Similarity. B. Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. original: G(2, 1), H(4, 1), I(2, 0), J(4, 0) image: Q(4, 2), R(8, 2), S(4, 0), T(8, 0) Since the figures are rectangles, their corresponding angles are congruent. Find and compare the ratios of corresponding sides. Answer: A. Graph the original figure and its dilated image. Then determine the scale factor of the dilation. original: B(–7, –2), A(5, –2), D(–7, 7) image: J(–3, 0), K(1, 0), L(–3, 3) 1 A. __ 2 1 B. __ 3 3 C. __ 2 3 D. __ 4 B. Graph the original figure and its dilated image. Then determine the scale factor of the dilation. original: A(4, 3), B(6, 3), C(4, 2), D(6, 2) image: E(6, 4), F(10, 4), G(6, 2), H(10, 2) A. 2 1 B. __ 3 C. 3 D. 4 7-6 Assignment Page 514, 6-13, 15