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the image AT'U'V'. e.) Identify the coordinates of AT'U'V'. T' (-3,2) T" v. (20) v. (5,-5) V' shape shape. (x,y) → ( 62 ) and Congruent figures have the same size Similar figures have the same but not necessarily the same f.) Using the image of AT'U'V' from example 4, perform an additional translation using the rule (x, y) → (x + 3, y-3). State the new coordinates of AT"U"V". Is this new image congruent or similar to the original figure? Size W Practice: 1.) b.) Graph and label the image after the translation. c.) Name the coordinates of the image. A' ·(1, 2) B. (4, 2) C² (2₁-1) D. (1₁-1) B' C' D' 2.) a.) Use arrow notation to write a rule for the given translation. (x,y) → (x+5₂ y+₁) a.) Use arrow notation to write a rule for the given translation. (x,y) → (x-3, y-2) b.) Graph and label the image after the translation. c.) Name the coordinates of the image. A⋅ (-2₁1) B⋅ (-1, 1) ∞ (0₁-1) D² (-1,-1) C' 3.) D G N D' FN y Other Transformation G' اينا E P4 X 4.) In questions 3 and 4 below, use arrow notation to write a rule that describes the translation shown on the graph. 64 right 5 units, up 1 unit Ay 4 tation: T(a,b) (x,y)= (x + a, y + b) A D le 12 -4 -2 'A' in 'DI NOTI left 3 units, down 2 units y 2- 10 (x, y) = (x +4₁ y +₁) 5.) MULTIPLE CHOICE: Write a description of the rule (x,y) → (x−7, y + 4). (a) translation 7 units to the right and 4 units up (b) translation 7 units to the left and 4 units down (c) translation 7 units to the right and 4 units down (d)) translation 7 units to the left and 4 units up B 10 CH D tex O c 2 A' AB LT N OD CX 2 4 E X BB tol F (x,y) → (x-4, y to) X Translations Homework 1.) Plot ARST with coordinates R(2,3), S(-3,5), and T(0,6). Find the coordinates of the vertices of AR'S'T', the image of ARST under the translation T(3,-5). R' (5,-2) s (0,0) T (3, 1) S' 2.) What is the image of point R (1, 3) under T(-2, 2)?. 3.) What is the image of point R (2, 2) under T(3, -1)? (-,5). (15,1) Y 4.) The image of point K (-1, 3) is K' (5, 7). What is the vector by which K was translated? 5.) The image of point K (-2, -5) is K' (1, 3). What is the vector by which K was translated? -8-7-6-5-4-3-2-1 8.) What is the image of point Q (-2, -7) under T(6, 4)?. (4,-3) 5 -2 9.) The image of point P (6, -4) is P' (-2, 5). What is the vector by which P was translated? -4 -6 8 23456 6.) If the image of A (4, 1) is A' (-2, 4) under a translation, what are the coordinates of the image of B (0, -6) (-6,3) under the same translation? -6₂-3) 7.) If the image of A (-1, 2) is A' (3, -3) under a translation, what are the coordinates of the image of B(1, -3) under the same translation? (5-8) Use for #1 only (6,4) (3,8) (-8,9) 'R' 10.) If the image of R(-9, 2) is R'(-1, -5) under a translation, what are the coordinates of the image of S(4, -8) (3₁7) (12,-1) under the same translation? Name: This line is called the_ A reflection is a transformation which EXAMPLE 1: 4ABC is being reflected over the x-axis. Draw and label the image 4A'B'C'. We can use an arrow to describe this reflection. A Honors Geometry: Reflection Notes AABCAA'B'C' What are the coordinates of: (1₁-3) ·(1,3) B ·(3,0) (3,0) c(4,-2) c(4,2) flips line of reflection A' EXAMPLE 2: B' Date: AABC is reflected over the y-axis. Draw the image AA'B'C'. Are these figures congruent or similar? Explain how you know. size & shape What are the coordinates of: A (1₁-3) → A' (-4,-3) B (3,0) → B. (-3,0) B' the figure over a c : (4₁-2) c(-4,-2) 2 given 2 -2 Period: 49² 2 3 Write a general rule for an x-axis reflection: x. -y, rx-axis (x,y) → ( congruent - same - Cand similar, all are similar -2 line. xt To C B 1x C Write a general rule for a y-axis reflection: Fy-axis(x, y) → (_X_,. .y EXAMPLE 3: a.) Draw AJKL which has coordinates J (0,2), K (3,4), and L (5,1). b.) Draw the image AJ'K'L' after a reflection of AJKL over the x-axis. c.) List the coordinates of J'K'L'. J (0, 2) → K (3,4) L (5, 1) >> → d.) Draw the image AJ"K"L" after a reflection of AJ'K'L' over the y-axis. e.) List the coordinates of J''K'L". J" J. (0, -2) x-(3-4)→ EXAMPLE 4: (-2) *-(-3,-4) K 1·(5, -1)→ ₁(-5, -1) L B J. (0, -2) J' C K. K' (3,-4) 1. (5,-1) a.) Draw AABC which has coordinates A(0,1), B(3,4), and C(5,1). b.) Draw the image 4A'B'C' after a reflection of AABC over x =-1. c.) List the coordinates of A'B'C'. A (0, 1) → (3,4) → f.) Describe a different combination of two reflections that would move AJKL to AJ"K"L". reflect orer y first then x g.) Is this new image congruent or similar to the original figure? congment (and similes) (5, 1) → 1 (=2₁1) A B. (-5,4) B' c.(-7,1) 8 -7 DU O 6 2 X=-1 81 6 4 2 -8-7-6-5-4-3-2-1 244 20 -4 -6 B 12345678 EXAMPLE 5: a.) Draw AABC which has coordinates A(0,1), B(3,4), and C(5,1). b.) Draw the image 4A'B'C' after a reflection of AABC over y = -2. c.) List the coordinates of A'B'C'. A (0, 1) ➜ A' B с L (3,4) → M (5, 1) → EXAMPLE 6: Draw the line of reflection which caused rectangle KLMN to reflect onto rectangle K'L'M'N'. What is the equation of the line of reflection? KK' -+ NN' B' (3₁-8) c² (5₁-5) C' L' (0₁-5) M₁ X=0 EXAMPLE 8: Quadrilateral CDEF is plotted on the grid to the right. a.) On the graph, draw the reflection of polygon CDEF over the x-axis. Label the image C'D'E'F'. down 4 (0-4) reflect over x = 0 b.) Now create polygon C"D"E"F" by translating polygon C'D'E'F' three units to the left and up two units. What will be the coordinates of point C"? (4,0) EXAMPLE 9: Describe how you could move shape 2 to exactly match shape 2' by using one translation and one reflection. (y-axis) (or use the reverse order)" 3 -4 -2 CHEL 1 EXAMPLE 7: Draw the line of reflection which caused triangle ABC to reflect onto triangle A'B'C'. What is the equation of the line of reflection? A A' 2 3 4 C C -4 -3-2-1 AV 3 FR 3 B O B' 6 Ay 4 24 -6 O FC 2 Air 3 11' 2345678 D 27 1 C B C' /1 X CI xix →→y=-2 ·y=-1 EXAMPLE 10: a.) On the coordinate plane graph the line y = x b.) Graph ARST with coordinates R(1,4), S(1,8), T(3,8). * reflect over y=x c.) What are the coordinates of R'S'T'? 나, R (1,4) R'(. S (1,8) S (8 T (3,8)→ T'(8. 3 1 3 Write a general rule for a y = x1 eflection: ry=x(x,y) → EXAMPLE 11: a.) On the coordinate plane graph the line y = -x Point Reflections: I b.) Graph ABUG with coordinates B(-1,5), U(1,8), G(4,4). *reflect over y=-x c.) What are the coordinates of B'U'G'? B (-1, 5) → B’ (5, 1) U (1,8) → U’ (8,-1) G (4,4)→ G²(4,4 A' Write a general rule for a y = -x reflection: ry=-x (x,y) → ( C ∙y. 10-8-6-4-2 -10 x B yo Fort -6 *) *_y.__x 10 6 4 -2² 4 -6 -8 10 -4 -2 G₁ R 2 4 2 -2 -4 -6 0 -8 -10 4 b 2 4 18 General Rules: From each figure point count the non-reduced rise/run to the point you are reflecting over. Continue the slope one time and plot image of figure point. 10 y=x 6 8 10 у=-ху Reflections Homework 1.) What are the coordinates of point (2,-3) after it is reflected over the x-axis? (2, 3) 2.) Point (-2, 3) is reflected in the y-axis. In which quadrant does its image lie? I 3.) The vertices of triangle ABC are A (1,6), B(4,3) and C(2,-1). Graph triangle ABC and reflect it over the y-axis and label it A'B'C'. Be sure to list the coordinates of A'B'C'. 4.) The vertices of quadrilateral ABCD are A(1,3), B(2,1), C(7,1) and D(5,5). Graph ABCD and reflect it over the x-axis and label it A'B'C'D'. Be sure to list the coordinates of A'B'C'D'. B' B. (-4,3) c-(-2,-1) name points in reflection 4·(1-3) A B C' 3' (2₁-1) .. (7,-1) (5₁-5) D' B₁ -8-7-6-5-4-3 Alop CL -8-7-1 -1 T' -8-7-6-5-4-3-2-1 2 8 81 6 4 2 -2 -6 -8 A function, f, is defined by the set {(2,3), (4,7), (-1,5)}. If f is reflected in the line y point will be in the reflection? (3₁2) (7,4) (5,-1) 2-3 IN BIS A' -2 6.) The vertices of triangle RAT are R(-3,0), A(-4,3) and T(-3,-4). Graph triangle RAT and reflect it over the line y = x and label it R'A'T'. Be sure to list the coordinates of R'A'T'. R' (0, -3) усха 1(3,-4) T(-4,-3) -4 B LO CO 3 4 5 6 7 8 -6 'D' = x, which 3 4 5 6 7 8 A¹ 7.) What are the coordinates of point P, the image of point (3,-4) after a reflection in the line y = -x? (4,-3) 8.) The vertices of quadrilateral RECT are R(-6,-3), E(-5,1), C(-1,-3) and T(-1,1). Graph RECT and reflect it over the line y=-x and label it R'E'C'T'. Be sure to list the coordinates of R'E'C'T'. R. 3,6 E' (1,5) T' (1) (2) c²(3,1) (-) D 9.) The graph to the right shows the relationship between kinetic energy, y, and velocity, x. 10.) Which graph is the reflection of this graph in the line y = x ? B¹ (2 C' (3 D'(2 ) ) y E₁ (3) ) 14 -8-7-6-5-4-3-2-N2 3 4 5 6 7 8 102 2 -4 -6 لا y E-b (4) 11.) On the accompanying set of axes, draw the reflection of ABCD in the y-axis. Label and state the coordinates of the reflected figure. -2 ان X ² y=x -2) -4) 12.) A point P has coordinates (-4, 7). What are its new coordinates after reflecting point P in the x- axis? (-4,-7) у=-х 13.) A point P has coordinates (-1, -8). What are its new coordinates after reflecting point P in the y- axis? (1₁-8) 14.) What is the reflection of (-2, 3) in the line y=-x? (-3,2) Key turning Figures can be turned in two directions: clockwise Name: Rotation: The two "D's" of rotations: EXAMPLE 1: Starting Point A (1,4) direction (which way) After a rotation has been performed, is the image going to be similar or congruent? B (5,2) C (2,0) Honors Geometry: Rotation Notes a figure about a fixed point degree (how far) B" "B" A" The figure below shows counterclockwise rotations of 90°, 180°, 270° and 360°. Use the figure to complete the table. 90° (A'...) Rotation CC (-4,1) (-2,5) A Date: C B and and 180° (A"...) Rotation CC Period: counterclockwise R90° (x, y) R180° (x, y) R270° (x, y) R360° (x, y) congruent (4 Skimilas) 270° (A"...) Rotation CC Below "CC" means Counterclockwise 360° (A...) Rotation CC (-1,-4) (4, -1) (1,4) (-5₁-2) (2-5) (5,2) (0₁2) (-2,0) (0₁-2) (2,0) Complete each rule for finding the image of any point (x, y) under the given rotation. Standard rotation is always counterclockwise. 90° rotation about the origin: (-y. x) 180° rotation about the origin: 270° rotation about the origin: 360° rotation about the origin: (y₁-x) -cty EXAMPLE 2: Draw the final image created by rotating triangle RST 270° counterclockwise about the origin. Be sure to list the coordinates of R'S'T'. R(-4,4) S(-2,3) T(-3,1) R' 4,9 s(3, 2) r(1, 3) ·(4₁-4) R' S' (3,-2 T. (1, -3) T' R R' (3,2) s'(4, 5) T. (8,4) EXAMPLE 3: Draw the final image created by reflecting triangle RST in the x-axis and then rotating the image 90° counterclockwise about the origin. List the coordinates of R'S'T'. Ro (-4,-4) S° (-2, -3) T° (-3,-1) ハ -4 -2 EXAMPLE 4: a.) Graph Triangle RST with vertices R(2, 3), S (5, 4), and T (4, 8). T' R Tis S 2 -2 -4 2 S -4 O 2 +2 y Are the final images in examples 2 and 3 the same? Why or why not? Not in same location, but same size. All side length 2. 8-7-6-5-4-3-2-1 12345 10 2 b.) Using the rule for a rotation of 90° counterclockwise, graph Triangle R'S'T' on the graph below and write the new coordinates. -** 2 S 4 X EXAMPLE 5: Quadrilateral ABCD is plotted on the grid below. a.) On the graph, draw the image of quadrilateral ABCD after a counterclockwise rotation of 180° about the origin. Label the image A'B'C'D'. b.) Explain how the coordinates of A changed to the coordinates of A'. A(5,6) B (10,6 C(8.1) 1 D(3,1) 180° either direction (x,y) → (x, y) EXAMPLE 6: Explain how each of these figures was rotated. a.) b.) F A(-4,2) ¹². A1 (-5,6) B₁ (-10,6) O 01 (8,4) D'(-3,1) B(1,3) C (-21) c(-1,2) D' D(-32) D. (2₁3) (-2,3) Rotation B. (-3,-1) A(-2,-4) B' B(-3,1) c(-1,2) D(+2, -3) 12-11-10 1 A H' D O 12 EXAMPLE 7: Draw the final image created by rotating polygon ABCD 90° counterclockwise about the origin and then reflecting the image in the x-axis. List the coordinates of A'B'C'D'. 3. 3 11 H Ay F D B 3 C 270° CC ок 90⁰ c X Rotations Homework: 1.) What are the coordinates of (3, -2) under a 90° counterclockwise rotation about the origin? (2,3) 2.) What are the coordinates of (-5, 4) under a 180° counterclockwise rotation about the origin? 5,-4) 3.) What are the coordinates of (3, 2) under a 90° clockwise rotation about the origin? (2,-3) 4.) Rotating 90° clockwise about the origin is equivalent to rotating about the origin. 5.) Rotating 270° clockwise about the origin is equivalent to rotating_90_counterclockwise about the origin. 6.) Rotating 180° clockwise about the origin is equivalent to rotating 180 counterclockwise about the origin. List all coordinates. R.1-8) R² S' 5 (69) T' (3-4 7.) Point A (3, 6) is rotated 270° counterclockwise about the origin, what is the coordinate of A' ? -3) 8.) a.) Graph ARST with coordinates R(8,1), S(9,6), T(4,3). b.) Graph AR'S'T' the image ARST under a rotation of 270 degrees. c.) Graph AR"S"T" the image AR'S'T' under T.3,2 d.) Graph AR""S""T" the image AR"S"T" under a reflection in the x-axis. R' R. (-2, 6) S--(3₁-7) T: (0₁-2) T" (-26) ·(317) T(0,2) 270⁰ R" S -10 -8 -6 -4 -2 10 8 6 #N 4 2k -26T" 44 counterclockwise R"-6 -8 "R' -10 O 4 6 LO SII 8 10 st Name: Honors Geometry: Dilation Notes Dilation: transformation that produces an image that is the same shape as the original but NOT same size A dilation is ● similar to the original figure. Dilations are centered around the origin (0, 0), unless otherwise stated. Scale factor - is ratio. Targer If the scale factor is greater than 1, the figure becomes If the scale factor is between 0 and 1, the figure becomes smaller R' image length pre - image length General Rule: D (k) (x, y) = (k.x, k.y) EXAMPLE 1: If the scale factor is 3, how would you write the rule? A' which is a Date: EXAMPLE 2: Triangle ABC has vertices A (0, 2), B (4, 4), and C (-1, 4). What are the vertices of its image with a scale factor of 4? ·(0,8) B. (16,16) B' a.) Find and graph the coordinates of the image. P. (-1,2) –1, (2₁-1) 0 (2, 2) -(-2,-2) S' EXAMPLE 3: Quadrilateral PQRS has vertices P (-2, 4), Q (4, 4), R (4, -2), and S (-4,-4). It is dilated by a scale factor of 2. b.) Demonstrate these quadrilaterals are similar by comparing the ratios of the side lengths. 2:1 2:1 2:1 • D₂ (x,y) = (3x₁ ³y) c. (-4,16) C' -6 -4 S PG=6 P'Q' = 3 QR=6 Q¹R¹ = 3 RS=168 R'S' 7 PS= 168 Pis = 117 c.) What do you notice about the angle measurements of the two figures? They remain the same. (=) 6 2 -4 Period: 32 lo y 6 Q! Q 6 X 5 EXAMPLE 4: If the scale factor is how would you write the general rule: enlargement 2 Is this an enlargement or a reduction? EXAMPLE 5: Quadrilateral A'B'C'D' is a dilation of quadrilateral ABCD. a.) Find the scale factor. A (0,6) A' (0,2) X(-1,-1) 9(1, 2) Z(1,1) 6• K = 2 K = 2 K = 3/32 b.) Is this dilation an enlargement or a reduction? reduction I Delay) (Fix, 3y) Y TEN JUNE 2 4 4 b 2 EXAMPLE 6: Triangle XYZ is graphed below. Draw and label Triangle X'Y'Z' after a dilation using a scale factor of two. X' x² (-2,-2) x² (-2,4) Y' z² (2₁2) Z' B B # 6 D 6 Dilations Homework: 1.) If A(1, 3) is dilated by a factor of 4, what are the coordinates of A'? (4,12) 2.) Under a dilation with a scale factor of k, the image of point (-2,6) is (-8, 24). What is the value of k? K = 4 3.) Under a dilation of constant k, the image of point (0, 3) is (0, -6). What are the coordinates of (-2, 5) under the same dilation? K=-2 3K=-6 K=-2 (4,-10) 4.) If Z(4, -16) is dilated by a factor of ½, what are coordinates of Z²? (2, -8) 5.) Under a dilation of constant k, the image of point (-5, 16) is (2.5, -8):. What are the coordinates of (-2,5) under the same dilation. (1,-2.s) -5K=2.5 K=2.5 k==== 6.) a.) Graph triangle ABC with A (2, 3), B(-1, 1) and C (1, -3). b.) Reflect triangle ABC over the x-axis and label it A'B'C'. c.) Dilate triangle A'B'C' by a constant of 3 and label it A"B"C". d.) Translate triangle A"B"C" by T(-3, 1) and label it A"""B""C"". ·(2₁-3) (-1,-1) c (1,3 B' C' A' (6₁-9) (-3,-3) .. (3,9) A" -2K = -8. K=4 ; B" C" ·(3,-8) B... (-6₁-2) B" (0, 10) C"" -00 500 P B 10% CA < 10 -8 -6 -4 -2 3 do ANNA O -4 -B -8 to Al Alli 14 6 8 10 Z Transformation Practice: 1.) Find the coordinates of the image ABCD with vertices A(0, 0), B(0, 3), C(3, 3), and D(3, 0) after a 4 dilation with a scale factor of A' (0,0) B'(0,4) 2.) Find the coordinates and graph the image of quadrilateral WXYZ after a dilation about the origin with a scale factor of 1. If the image was rotated 90° clockwise, what would be the coordinates 2 of X"? (2,-1) (0,2) Xx'. (4,2) (4₁-1) r. (2, 1) Y' 2² (2, -1/2) area (0, 1) w²(4, -1/2) x (10) X" Q ratio : 2 3.) AQ'P'R' is a dilation of AQPR. Find the scale factor. Is it an enlargement of a reduction? Q' 4 C¹ (²4,4) D'(4,0) P = P¹ and ABC. A' (-3,-3) B₁ (6₁-3) 0² (1.5,1.5) B' 20.259 R R' W 4.) A triangle has coordinates A(-2, -2), B(4, -2), and C(1, 1). Graph its image A'B'C' after a dilation with scale factor Give the coordinates of A'B'C', and the ratio of the areas of the figures A'B'C' Area ABC= 1/2 (6)(3) = 9u² Area A'B'C' = 3 (9) (4.5) = 20.25u² O 9:4 which is 3²: 2² X Y 4 B' 45 5.) A graphic artist tried to translate a copy of the original school bus drawing below, but he accidentally left one of the windows behind. a.) Make a table showing the coordinates of the vertices of the left window of the original bus and the coordinates of the vertices that this window should have in the image. (13, 12) (-10,-3) (-8, -3) (-10,-5). (-8,-5)- 6.) Draw and label the image of the figure after a reflection over the x-axis. Ay -3 -3 A b.)Describe the translation so someone else could start with the drawing of the original bus in the bottom left-hand corner and draw the correct image shown in the upper right-hand corner. More all points right 13 units and up 3 (5,9) (37) (517) 1 O 3 8.) Based on the given drawing, determine the specific transformation. X reflection over x=-3 3 7.) Draw and label the image of the figure after a reflection over the line y = 1. Ay 3 DY' School 3 A' D H O OHH (†) 23 3 School Bus X 9.) Based on the given drawing, determine the specific transformation. 12 units > y=1 reflection over x-axis 10.) What single transformation is equivalent to a reflection in the y-axis, followed by a reflection in the x-axis, followed by a reflection in the y-axis? A A B 11.) What single transformation is equivalent to a reflection in the x-axis, followed by a reflection in the y-axis? ΤΑ A's b.) What were the coordinates of the drummer's final position? (3-4,4-3) = A(-1,-4) B'(-1,0) c'(-4,0) (x₁-y) 180° rotation about origin (-1, 1), B 5 K = = = 141 20 13.) Which translation below is NOT described by the rule (x,y) → (x+2, y − 3)? a.) (3,-2)→→ (5,-5) b.) (− 4,1) → (− 2,−2) c.) (0,4)→ (2,1) d.) (1,-5) → (3,-2) reflection over x-axis 12.) The width of a picture is 20 cm. Using a color copier, you reduce the width of the picture to 5 cm. What is the scale factor of the dilation? 20K=5 K= A (1,4) B (1,0) C(4,0) C 14.) At the half-time show, a marching band marched in formation. The lead drummer started at a point with coordinates (3, 4) and moved 3 steps down and 4 steps left. a.) Write a rule to describe the translation T(-4,-3) or (xx,y) → (x-4, Y-3) (x-4,4-3)