9. Yes; translate so that the centers align, and then dilate using the ratio of radii to map one circle to another.

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Roy Beasley
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1 9. Yes; translate so that the centers align, and then dilate using the ratio of radii to map one circle to another. 10. Keegan dilated using A as the center of dilation instead of the origin. Point A should rotate 90 to ( 1, 1) and then dilate by 2 to ( 2, 2) if the dilation is centered on the origin like it should be. 11. m A = 103, m B = 100, m C = 80, m D = 77 ; Rigid motion preserves angle measurements, therefore A A', B B' C C' and D D'. Use the fact that the angle of a straight line is 180 to find m B and m D, where m D also needs the fact that altitude angles are equal and the angles of a triangle sum to 180. Use the fact that the angle between parallel lines is 180 to find m A and m C. 180 = m D ( ) 77 = m D m A + m D = 180 m A + (77 ) = 180 m A = = m B = m B m B + m C = 180 (100 ) + m C = 180 m C = Yes; use rigid motions to map a vertex of one square onto a vertex of the other square with the sides aligned. Then dilate so that the side length is the same. Since the interior angles and the ratio of the height to width of a square are, by definition, the same for all squares, all squares are similar. 13. In order to show whether a composition of a dilation and a translation can be performed in either order and have the same result, we test the validity of the following statement. (D k T a,b )(x, y) = (T a,b D k )(x, y) (D k T a,b ) (x, y) = D k (x + a, y + b) = (kx + ka, ky + kb) (T a,b D k ) (x, y) = T a,b (kx, ky) = (kx + a, ky + b) Thus, the statement (D k T a,b )(x, y) = (T a,b D k )(x, y) ; the order of a composition of a dilation and a translation determines the result.

2 14. a. ABC DEF; The segment connecting any two midpoints in a triangle is parallel to and half the length of the opposite side. Because of properties of parallel lines and transversals, the angles of the two triangles can be shown to be congruent. The side lengths of DE F are half the lengths of ABC, so the scale factor is 0.5. b. ABC is reflected across AC, dilated with center of dilation B and scale factor of 0.5, and translated. 15. X (14, 7), Y (11, 2.5), Z (2, 0.5) Preimage Dilated (1.5) Translated 5, 4 X(6, 2) (9, 3) (14, 7) Y (4, 1) (6, 1.5) (11, 2.5) Z( 2, 3) ( 3, 4.5) (2, 0.5) 16. L (1, 2), M (2, 2), N (2, 2), P' (1, 2) Preimage Dilated (0.5) Reflected (x-axis) L(2, 4) (1, 2) (1, 2) M(4, 4) (2, 2) (2, 2) N(4, 4) (2, 2) (2, 2) P(2, 4) (1, 2) (1, 2)

3 17. P ( 6, 2), Q ( 4, 8), R ( 10, 0) ; Preimage Rotated (270 ) Dilated (2) Reflected (y-axis) P(1, 3) (3, 1) (6, 2) ( 6, 2) Q(4, 2) (2, 4) (4, 8) ( 4, 8) R(0, 5) (5, 0) (10, 0) ( 10, 0) 18. A (0.5, 1.5), B (0, 0), C ( 1.25, 2), D ( 0.5, 2.5) Preimage Reflected (x-axis) Dilated (0.25) A(2, 6) (2, 6) (0.5, 1.5) B(0, 0) (0, 0) (0, 0) C( 5, 8) ( 5, 8) ( 1.25, 2) D( 2, 10) ( 2, 10) ( 0.5, 2.5) 19. Answers may vary. Sample: translation down 2 units and right 4 units, and then dilation with the point P' as center and scale factor ; 20. Answers may vary. Sample: reflection across x = 3, dilation with center at A' and scale factor 2; ABC ZY X 21. Answers may vary. Sample: rotation 180 and dilation to decrease the image size; HJKL ABCD Yes; it appears you can scale the figure on the left by and map the image onto to the figure on the right. 2 PQRS DABC 23. No; the bases of the trapezoids are congruent but the top trapezoid appears to be shorter in height than the bottom trapezoid. 2

4 24. He can use the 3 ft by 4 ft paper if he rotates it so that the width is 4 ft and the height is 3 ft because the side lengths are proportional to his cutout. The 180 cm by cm paper does not have side lengths that are proportional to his cutout. He can use it if he cuts it down to 160 cm by cm (180 20) =, = = m ² ; 5.5 m is represented by 11 sections on the grid, so each grid line represents = 0.5 m. The scrim is 10 grid sections tall, or = 5 m = The original area of the floor is 6 2 = 36 in. 2. Use the new area 25 in. ² to solve for the scale factor of the new dollhouse: A k 2 = A (36) k 2 = 25 k 2 = k = Multiply the original dollhouse pattern dimensions by the scale factor to find the dimensions for the smaller dollhouse, and sketch.

5 27. Preimage Dilated (2) Translated 1, 3 A(5, 2) (10, 4) (11, 7) B( 1, 3) ( 2, 6) ( 1, 3) C(3, 2) (6, 4) (7, 1) 28. (D) R x axis (8, 2) = (8, 2 ( 1)) = (8, 2) D 4 (8, 2) = (8 4, 2 4) = (32, 8) 29. Part A Check students work. Part B Check students work. Part C Yes; all figures are similar to each other, so there is a similarity transformation that maps any figure to any other figure.

1 What is the solution of the system of equations graphed below? y = 2x + 1

1 What is the solution of the system of equations graphed below? y = 2x + 1 3 As shown in the diagram below, when hexagon ABCDEF is reflected over line m, the image is hexagon A'B'C'D'E'F'. y = x 2 + 2x