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 9 / 154 Slide 10 / 154 Transformations 1 Does the transformation appear to be a rigid motion? Explain. A transformation maps every point of a figure onto its image A Yes, it preserves the distance between consecutive and may be described using arrow notation ( ). points. Prime notation ( ' ) is sometimes used to identify image points. B No, it does not preserve the distance between consecutive points. In the diagram below, A' is the image of A . A' A # ABC # A'B'C' # ABC maps onto # A'B'C' B B' C C' Preimage Image 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 11 / 154 Slide 12 / 154 2 Does the transformation appear to be a rigid motion? 3 Which transformation is not a rigid motion? Explain. A Yes, distances are preserved. A Reflection B Yes, angle measures are preserved. B Translation C Both A and B. D No, distance are not preserved. C Rotation D Dilation Preimage Image

Slide 13 / 154 Slide 14 / 154 4 Which transformation is demonstrated? 5 Which translation is demonstrated? A Reflection A Reflection B Translation B Translation C Rotation C Rotation D Dilation D Dilation Slide 15 / 154 Slide 16 / 154 6 Which transformation is demonstrated? A Reflection B Translation Translations C Rotation D Dilation Return to Table of Contents Slide 17 / 154 Slide 18 / 154 Translations Translations A translation is a transformation that maps all points of a figure Write the translation that maps the same distance in the same direction. ABC onto A'B'C' as T( ABC) = A'B'C' B' A translation is a rigid motion with the following properties: B AA' = BB' = CC' A' A C' AB = A'B', BC = B'C', AC = A'C' C m<A = m<A', m<B = m<B', m<C = m<C'

Slide 19 / 154 Slide 20 / 154 Translations in the Coordinate Plane Finding the Image of a Translation B is translated 9 units right What are the vertices of T <-2, 5> ( DEF)? and 4 units down. Graph the image of DEF. Each ( x, y ) pair in ABCD is mapped to ( x + 9, y - 4). B A D' ( ) E' ( ) D You can use the function notation F' ( ) A' B' T <9,-4> ( ABCD ) = A'B'C'D' D C to describe the translation. Draw DD', EE' and FF '. E F C' D' What relationships exist among these three segments? How do you know? Slide 21 / 154 Slide 22 / 154 Writing a Translation Rule 7 In the diagram, ΔA'B'C' is an image of ΔABC. Which rule describes the translation? Write a translation rule that maps PQRS P'Q'R'S'. A T <-5,-3> ( ABC) P B T <5,3> ( ABC) S C T <-3,-5> ( ABC) P' Q D T <3,5> ( ABC) S' R Q' R' Slide 23 / 154 Slide 24 / 154 8 If T <4,-6> (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 T <4,-6> (J'K'L'M') A (-6,-4) B T <6,-4> (J'K'L'M') B (-3,2) C T <6,4> (J'K'L'M') C (-3,9) D T <-4,6> (J'K'L'M') D (-4,13) 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 28 / 154 Reflection Reflection When reflecting a figure, reflect the vertices and then draw the sides. Reflect WXYZ over line s. Label the vertices of the image. Reflect ABCD over line r. Label the vertices of the image. Click here to r W Z see a video D A X Y B C s Hint: Turn page so line of symmetry is vertical Slide 29 / 154 Slide 30 / 154 Reflection 10 Which point represents the reflection of X? Reflect MNP over line t. Label the vertices of the image. X A point A Where is the image of N? Why? A B point B M B C point C C D point D N P E None of the above D t

Slide 31 / 154 Slide 32 / 154 11 Which point represents the reflection of X? 12 Which point represents the reflection of X? D A A point A A point A X B point B B point B X B C C point C C point C A C D D point D D point D B E none of the above E none of the above Slide 33 / 154 Slide 34 / 154 13 Which point represents the reflection of D? 14 Is a reflection a rigid motion? A point A Yes A B point B No D C point C B C D point D E none of these Slide 35 / 154 Slide 36 / 154 Reflections in the Coordinate Plane Reflections in the Coordinate Plane Reflect A, B, & C over the y -axis. Since reflections are perpendicular to and equidistant from the How do the coordinates of each point change when the point is line of reflection, we can find the exact image of a point or a reflected over the y -axis? figure in the coordinate plane. Notation A R y-axis ( A ) = A' R y-axis ( B ) = B' R y-axis ( C ) = C' B C

Slide 37 / 154 Slide 38 / 154 Reflections in the Coordinate Plane Reflections in the Coordinate Plane Reflect A, B, C & D M over the line y = x . Reflect figure JKLM over B the x- axis. A Notation L K Notation R y=x ( A ) = A' R x-axis ( JKLM) = J'K'L'M' J C R y=x ( B ) = B' R y=x ( C ) = C' D R y=x ( D ) = D' How do the coordinates of each point change when the point is How do the coordinates of each point change when the point is reflected over the y -axis? reflected over the x -axis? Slide 39 / 154 Slide 40 / 154 Reflections in the Coordinate Plane Reflections in the Coordinate Plane Reflect quadrilateral A N MNPQ over y = -3 Reflect ABC over x = 2. Δ B M P C Notation Notation R x =2 ( ABC) = A'B'C' Δ Δ R Y=-3 ( MNPQ) = M'N'P'Q' Q *Hint: draw line of reflection first Slide 41 / 154 Slide 42 / 154 Find the Coordinates of Each Image 15 The point (4,2) reflected over the x-axis has an image of ______. 1. R x-axis ( A ) A (4,2) 2. R y-axis ( B ) B (-4,-2) F C A C (-4,2) 3. R y=1 ( C ) D (4,-2) 4. R x=-1 ( D ) E D B 5. R y=x ( E ) 6. R x=-2 ( F )

Slide 43 / 154 Slide 44 / 154 16 The point (4,2) reflected over the y-axis has an 17 B has coordinates (-3,0). What would be the coordinates image of _____. of B' if B is reflected over the line x = 1? A (4,2) A (-3,0) B (-4,-2) B (4,0) C (-4,2) C (-3,2) D (4,-2) D (5,0) Slide 45 / 154 Slide 46 / 154 Line of Symmetry 18 The point (4,2) reflected over the line y=2 has an image of _____. A line of symmetry is a line of reflection that divides a figure A (4,2) into 2 congruent halves. These 2 halves reflect onto each other. B (4,1) C (2,2) D (4,-2) Slide 47 / 154 Slide 48 / 154 Draw Lines of Symmetry Where Draw Lines of Symmetry Where Applicable Applicable N O M C A B R P Q D E F

Slide 49 / 154 Slide 50 / 154 Draw Lines of Symmetry Where 19 How many lines of symmetry does the following have? Applicable A one B two C three D none Slide 51 / 154 Slide 52 / 154 20 How many lines of symmetry does the following have? 21 How many lines of symmetry does the following have? A 10 A none B 2 B one C 100 C nine D infinitely many D infinitely many Slide 53 / 154 Slide 54 / 154 22 How many lines of symmetry does the following have? 23 How many lines of symmetry does the following have? A none A none J H B one B one C two C two D infinitely many D infinitely many