Slide 1 / 145 Slide 2 / 145 Geometry Transformations 2014-09-08 www.njctl.org Slide 3 / 145 Slide 4 / 145 Table of Contents click on the topic to go to that section Transformations Translations Reflections Rotations Transformations Composition of Transformations Congruence Transformations Dilations Similarity Transformations Return to Table of Contents Slide 5 / 145 Slide 6 / 145 In a transformation, the original figure is the preimage, and the A transformation of a geometric figure is a mapping that results in a resulting figure is the image. change in the position, shape, or size of the figure. In the examples below, the preimage is green and the image is pink. In the game of dominoes, you often move the dominoes by sliding them, turning them or flipping them. Each of these moves is a type of transformation. translation - slide rotation - turn reflection - flip
Slide 7 / 145 Slide 8 / 145 Some transformations (like the dominoes) preserve distance and A transformation maps every point of a figure onto its image angle measures. These transformations are called rigid motions. and may be described using arrow notation ( ). To preserve distance means that the distance between any two points Prime notation ( ' ) is sometimes used to identify image points. Answer of the image is the same as the distance between the corresponding points of the preimage. In the diagram below, A' is the image of A . A A' To preserve angles means that the angles of the image have the same measures as the corresponding angles in the preimage. △ ABC △ A'B'C' △ ABC maps onto △ A'B'C' B B' Rotation-turn Reflection- Flip Dilation - Size change Translation- slide C C' Note: You list the corresponding points of the preimage and image in the same order, just as you would for corresponding points in congruent figures Which of these is a rigid motion? or similar figures. Slide 9 / 145 Slide 10 / 145 1 Does the transformation appear to be a rigid motion? Explain. 2 Does the transformation appear to be a rigid motion? Explain. A Yes, it preserves the distance between consecutive points. A Yes, distances are preserved. B Yes, angle measures are preserved. Answer B No, it does not preserve the distance between consecutive C Both A and B. points. Answer D No, distance are not preserved. Preimage Image Preimage Image Slide 11 / 145 Slide 12 / 145 3 Which transformation is not a rigid motion? 4 Which transformation is demonstrated? A Reflection A Reflection B Translation Answer B Translation Answer C Rotation C Rotation D Dilation D Dilation
Slide 13 / 145 Slide 14 / 145 5 Which translation is demonstrated? 6 Which transformation is demonstrated? A Reflection A Reflection Answer B Translation B Translation Answer C Rotation C Rotation D Dilation D Dilation Slide 15 / 145 Slide 16 / 145 A translation is a transformation that maps all points of a figure the same distance in the same direction. Translations B' A translation is a rigid motion with the B following properties: A' - AA' = BB' = CC' A C' - AB = A'B', BC = B'C', AC = A'C' C - m<A = m<A', m<B = m<B', m<C = m<C' Return to Table of Contents You write the translation that maps ABC onto A'B'C' as T( ABC) = A'B'C' Slide 17 / 145 Slide 18 / 145 Translations in the Coordinate Plane Finding the Image of a Translation B is translated 9 What are the vertices of T <-2, 5> ( DEF)? Graph the image of DEF. units right and 4 units down. A B Answer D' ( ) Each ( x, y ) pair in ABCD is E' ( ) mapped to ( x + 9, y - 4). A' B' D F' ( ) D C You can use the function notation T <9, -4> ( ABCD ) = A'B'C'D' E to describe the translation. D' C' F Draw DD', EE' and FF '. What relationships exist among these three segments? How do you know?
Slide 19 / 145 Slide 20 / 145 Writing a Translation Rule 7 In the diagram, A'B'C' is an image of ABC. Which rule describes the translation? Write a translation rule that maps PQRS P'Q'R'S'. A Answer Answer P B S P' C Q S' R D Q' R' Slide 21 / 145 Slide 22 / 145 8 If (JKLM) = J'K'L'M', what translation maps J'K'L'M' 9 RSV has coordinates R(2,1), S(3,2), and V(2,6). A onto JKLM? translation maps point R to R' at (-4,8). What are the coordinates of S' for this translation? A A (-6,-4) Answer B Answer B (-3,2) C C (-3,9) D D (-4,13) E none of the above Slide 23 / 145 Slide 24 / 145 A reflection is a transformation of points over a line. This line is called the line of reflection. The result looks like the preimage was flipped over the line. The preimage and the image have opposite orientations. A B Reflections Properties B' C -If a point B is on line m , then the image of B is itself ( B = B'). Reflections Activity m C' Lab A' -If a point C is not on line m , then m (Click for link to lab) is the perpendicular bisector of CC' The preimage C and its image C' are equidistant from the Return to line of reflection. Table of Contents The reflection across m that maps ABC A'B'C' can be written as R m ( ABC) = A'B'C
Slide 25 / 145 Slide 26 / 145 Reflect WXYZ over line s. Label the vertices of the image. When reflecting a figure, reflect the vertices and then draw the sides. Reflect ABCD over line r. Label the vertices of the image. Z W r Answer Answer D A X Y Watch How! s B C HINT : Turn page so line of symmetry is vertical Slide 27 / 145 Slide 28 / 145 Reflect MNP over line t. Label the vertices of the image. 10 Which point represents the reflection of X? M X A point A A Answer B point B Answer B N P C point C t C D point D E None of the above D Where is the image of N? Why? Slide 29 / 145 Slide 30 / 145 11 Which point represents the reflection of X? 12 Which point represents the reflection of X? A point A A point A D A Answer Answer B point B B B point B X X C A C point C C point C C D B D point D D point D E none of the above E none of the above
Slide 31 / 145 Slide 32 / 145 13 Which point represents the reflection of D? 14 Is a reflection a rigid motion? A point A Yes B point B No A Answer Answer D C point C B C D point D E none of these Slide 33 / 145 Slide 34 / 145 Reflections in the Coordinate Plane Reflections in the Coordinate Plane Since reflections are perpendicular to and equidistant from the line of reflection, we can find the exact image of a point or a figure in the coordinate plane. M L Answer Answer Reflect figure Reflect A, B, & C over K JKLM over the the y -axis. x- axis. A J Notation R y- axis ( A ) = A' Notation R x-axis ( JKLM) = J'K'L'M' R y- axis ( B ) = B' B R y- axis ( C ) = C' C How do the coordinates of each point change when the point is reflected over the x -axis? How do the coordinates of each point change when the point is reflected over the y -axis? Slide 35 / 145 Slide 36 / 145 Reflections in the Coordinate Plane Reflections in the Coordinate Plane Reflect A, B, C & D over the line y = x . B A Reflect ABC over x = 2. A Answer Answer Notation B R y=x ( A ) = A' C Notation C R y=x ( B ) = B' R x =2 ( ABC ) = A'B'C' R y=x ( C ) = C' D HINT : Draw line of reflection first. R y=x ( D ) = D' HINT : Count the number of diagonals from the point to the line of reflection. How do the coordinates of each point change when the point is reflected over the y -axis?
Slide 37 / 145 Slide 38 / 145 Reflections in the Coordinate Plane Find the coordinates of each image. 1. R x -axis ( A ) Reflect quadrilateral F 2. R y -axis ( B ) MNPQ over y = -3 N C A Answer Answer 3. R y = 1 ( C ) M P 4. R x = - 1 ( D ) E Notation D Q R y =-3 ( MNPQ) = M'N'P'Q' B 5. R y = x ( E ) 6. R x = - 2 ( F ) Slide 39 / 145 Slide 40 / 145 15 The point (4,2) reflected over the x-axis has an 16 The point (4,2) reflected over the y-axis has an image of ______. image of _____. A (4,2) A (4,2) B (-4,-2) B (-4,-2) Answer Answer C (-4,2) C (-4,2) D (4,-2) D (4,-2) Slide 41 / 145 Slide 42 / 145 17 B has coordinates (-3,0). What would be the coordinates of 18 The point (4,2) reflected over the line y=2 has an B' if B is reflected over the line x = 1? image of _____. A (4,2) A (-3,0) B (4,1) B (4,0) Answer Answer C (2,2) C (-3,2) D (4,-2) D (5,0)
Slide 43 / 145 Slide 44 / 145 Line of Symmetry If lines of symmetry exist, draw all of them for the figure. A line of symmetry is a line of reflection that divides a figure into 2 congruent halves. These 2 halves reflect onto each A B C other. Answer E D F Slide 45 / 145 Slide 46 / 145 If lines of symmetry exist, draw all of them for the figure. If lines of symmetry exist, draw all of them for the figure. N O M Answer Answer P Q R Slide 47 / 145 Slide 48 / 145 19 How many lines of symmetry does the following have? 20 How many lines of symmetry does the following have? A one A 10 Answer B two B 2 Answer C three C 100 D none D infinitely many
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