Transformations Composition of Transformations Congruence - PDF document

Slide 1 / 154 Slide 2 / 154 Geometry Transformations 2014-09-08 www.njctl.org Slide 3 / 154 Slide 4 / 154 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 / 154 Slide 6 / 154 Transformations Transformations 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 reflection - flip rotation - turn

Slide 7 / 154 Slide 8 / 154 Transformations Transformations Which of these is a rigid motion? Some transformations (like the dominoes) preserve distance and angle measures. These transformations are called rigid motions. Translation- slide Rotation-turn To preserve distance means that the distance between any two points of the image is the same as the distance between the corresponding points of the preimage. To preserve angles means that the angles of the image have the same measures as the corresponding angles in the preimage. Reflection- Flip Dilation - Size change Slide 8 (Answer) / 154 Slide 9 / 154 Transformations Transformations A transformation maps every point of a figure onto its image Which of these is a rigid motion? and may be described using arrow notation ( ). Rotation-turn Prime notation ( ' ) is sometimes used to identify image points. Translation- slide In the diagram below, A' is the image of A . Answer Translation, A' reflection, A rotation are rigid transformations. # ABC # A'B'C' # ABC maps onto # A'B'C' Reflection- Flip Dilation - Size change B B' C C' [This object is a pull tab] 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 or similar figures. Slide 10 / 154 Slide 10 (Answer) / 154 1 Does the transformation appear to be a rigid motion? 1 Does the transformation appear to be a rigid motion? Explain. Explain. A Yes, it preserves the distance between consecutive A Yes, it preserves the distance between consecutive points. points. B No, it does not preserve the distance between B No, it does not preserve the distance between Answer consecutive points. consecutive points. B [This object is a pull tab] Preimage Image Preimage Image

Slide 11 / 154 Slide 11 (Answer) / 154 2 Does the transformation appear to be a rigid motion? 2 Does the transformation appear to be a rigid motion? Explain. Explain. A Yes, distances are preserved. A Yes, distances are preserved. B Yes, angle measures are preserved. B Yes, angle measures are preserved. Answer C Both A and B. C Both A and B. C D No, distance are not preserved. D No, distance are not preserved. Image Image Preimage Preimage [This object is a pull tab] Slide 12 / 154 Slide 12 (Answer) / 154 3 Which transformation is not a rigid motion? 3 Which transformation is not a rigid motion? A Reflection A Reflection B Translation B Translation Answer D C Rotation C Rotation D Dilation D Dilation [This object is a pull tab] Slide 13 / 154 Slide 13 (Answer) / 154 4 Which transformation is demonstrated? 4 Which transformation is demonstrated? A Reflection A Reflection B Translation B Translation Answer D C Rotation C Rotation D Dilation D Dilation [This object is a pull tab]

Slide 14 / 154 Slide 14 (Answer) / 154 5 Which translation is demonstrated? 5 Which translation is demonstrated? A Reflection A Reflection Answer B Translation B Translation A C Rotation C Rotation D Dilation D Dilation [This object is a pull tab] Slide 15 / 154 Slide 15 (Answer) / 154 6 Which transformation is demonstrated? 6 Which transformation is demonstrated? A Reflection A Reflection B Translation B Translation Answer C C Rotation C Rotation D Dilation D Dilation [This object is a pull tab] Slide 16 / 154 Slide 17 / 154 Translations A translation is a transformation that maps all points of a figure the same distance in the same direction. A translation is a rigid motion with the following properties: Translations AA' = BB' = CC' AB = A'B', BC = B'C', AC = A'C' m<A = m<A', m<B = m<B', m<C = m<C' Return to Table of Contents

Slide 18 / 154 Slide 19 / 154 Translations in the Coordinate Plane Translations B is translated 9 units right Write the translation that maps and 4 units down. ABC onto A'B'C' as T( ABC) = A'B'C' Each ( x, y ) pair in ABCD is mapped to ( x + 9, y - 4). B A B' B You can use the function notation A' B' A' T <9,-4> ( ABCD ) = A'B'C'D' A D C to describe the translation. C' C C' D' Slide 20 / 154 Slide 20 (Answer) / 154 Finding the Image of a Translation Finding the Image of a Translation What are the vertices of T <-2, 5> ( DEF)? What are the vertices of T <-2, 5> ( DEF)? D' (-5, 9), E'(-7, 4) F'(-1, 3) Graph the image of DEF. Graph the image of DEF. D' ( ) D' ( ) Answer E' ( ) E' ( ) D D F' ( ) F' ( ) Draw DD', EE' and FF '. Draw DD', EE' and FF '. E E Segments are same length, since a translation F moves all points the same distance. F [This object is a pull tab] What relationships exist among What relationships exist among these three segments? these three segments? How do you know? How do you know? Slide 21 / 154 Slide 21 (Answer) / 154 Writing a Translation Rule Writing a Translation Rule Write a translation rule that maps PQRS P'Q'R'S'. Write a translation rule that maps PQRS P'Q'R'S'. T <10, 5> (PQRS) = P'Q'R'S' Answer P P S S P' P' Q Q [This object is a pull tab] S' R S' R Q' Q' R' R'

Slide 22 / 154 Slide 22 (Answer) / 154 7 In the diagram, ΔA'B'C' is an image of ΔABC. Which rule 7 In the diagram, ΔA'B'C' is an image of ΔABC. Which rule describes the translation? describes the translation? A A T <-5,-3> ( ABC) T <-5,-3> ( ABC) Answer B B T <5,3> ( ABC) T <5,3> ( ABC) B C T <-3,-5> ( ABC) C T <-3,-5> ( ABC) T <3,5> ( ABC) T <3,5> ( ABC) D D [This object is a pull tab] Slide 23 / 154 Slide 23 (Answer) / 154 8 If T <4,-6> (JKLM) = J'K'L'M', what translation maps J'K'L'M' 8 If T <4,-6> (JKLM) = J'K'L'M', what translation maps J'K'L'M' onto JKLM? onto JKLM? A A T <4,-6> (J'K'L'M') T <4,-6> (J'K'L'M') Answer B B T <6,-4> (J'K'L'M') T <6,-4> (J'K'L'M') D C C T <6,4> (J'K'L'M') T <6,4> (J'K'L'M') D T <-4,6> (J'K'L'M') D T <-4,6> (J'K'L'M') [This object is a pull tab] Slide 24 / 154 Slide 24 (Answer) / 154 9 RSV has coordinates R(2,1), S(3,2), and V(2,6). A 9 RSV has coordinates R(2,1), S(3,2), and V(2,6). A translation maps point R to R' at (-4,8). What are the translation maps point R to R' at (-4,8). What are the coordinates of S' for this translation? coordinates of S' for this translation? A (-6,-4) A (-6,-4) Answer B (-3,2) B (-3,2) C C (-3,9) C (-3,9) D (-4,13) D (-4,13) [This object is a pull tab] E none of the above E none of the above

Slide 25 / 154 Slide 26 / 154 Reflection 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 Reflections If a point B is on line m , then the image of B is itself ( B = B'). B Reflections Activity Lab B' C (Click for link to lab) If a point C is not on line m , then m is the perpendicular bisector of CC' A' m C' Return to The reflection across m that maps ABC A'B'C' can be written Δ Δ Table of as R m ( ABC) = A'B'C Δ Δ Contents Slide 27 / 154 Slide 27 (Answer) / 154 Reflection Reflection When reflecting a figure, reflect the vertices and then draw the sides. When reflecting a figure, reflect the vertices and then draw the sides. Reflect ABCD over line r. Label the vertices of the image. Reflect ABCD over line r. Label the vertices of the image. Click here to Click here to r r see a video see a video Answer D D A A [This object is a pull tab] B B C C Slide 28 / 154 Slide 28 (Answer) / 154 Reflection Reflection Reflect WXYZ over line s. Label the vertices of the image. Reflect WXYZ over line s. Label the vertices of the image. W W Z Z Answer X Y X Y [This object is a pull tab] s s Hint: Turn page so line of symmetry is vertical Hint: Turn page so line of symmetry is vertical