Geometry Transformations 2014-09-08 www.njctl.org Slide 3 / 154 - PDF document

Slide 1 / 154 Slide 2 / 154 Geometry Transformations 2014-09-08 www.njctl.org Slide 3 / 154 Table of Contents click on the topic to go to that section Transformations · Translations · Reflections · Rotations · Composition of Transformations · Congruence Transformations · Dilations · Similarity Transformations ·

Slide 4 / 154 Transformations Return to Table of Contents Slide 5 / 154 Transformations A transformation of a geometric figure is a mapping that results in a change in the position, shape, or size of the figure. 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 6 / 154 Transformations In a transformation, the original figure is the preimage, and the resulting figure is the image. In the examples below, the preimage is green and the image is pink.

Slide 7 / 154 Transformations Some transformations (like the dominoes) preserve distance and angle measures. These transformations are called rigid motions. 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. Slide 8 / 154 Transformations Which of these is a rigid motion? Translation- slide Rotation-turn Reflection- Flip Dilation - Size change Slide 9 / 154 Transformations A transformation maps every point of a figure onto its image and may be described using arrow notation ( ). Prime notation ( ' ) is sometimes used to identify image 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' 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 1 Does the transformation appear to be a rigid motion? Explain. A Yes, it preserves the distance between consecutive points. B No, it does not preserve the distance between consecutive points. Preimage Image Slide 11 / 154 2 Does the transformation appear to be a rigid motion? Explain. A Yes, distances are preserved. B Yes, angle measures are preserved. C Both A and B. D No, distance are not preserved. Image Preimage Slide 12 / 154 3 Which transformation is not a rigid motion? A Reflection B Translation C Rotation D Dilation

Slide 13 / 154 4 Which transformation is demonstrated? A Reflection B Translation C Rotation D Dilation Slide 14 / 154 5 Which translation is demonstrated? A Reflection B Translation C Rotation D Dilation Slide 15 / 154 6 Which transformation is demonstrated? A Reflection B Translation C Rotation D Dilation

Slide 16 / 154 Translations Return to Table of Contents 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: AA' = BB' = CC' AB = A'B', BC = B'C', AC = A'C' m<A = m<A', m<B = m<B', m<C = m<C' Slide 18 / 154 Translations Write the translation that maps ABC onto A'B'C' as T( ABC) = A'B'C' B' B A' A C' C

Slide 19 / 154 Translations in the Coordinate Plane B is translated 9 units right and 4 units down. Each ( x, y ) pair in ABCD is mapped to ( x + 9, y - 4). A B You can use the function notation A' B' T <9,-4> ( ABCD ) = A'B'C'D' D C to describe the translation. C' D' Slide 20 / 154 Finding the Image of a Translation What are the vertices of T <-2, 5> ( DEF)? Graph the image of DEF. D' ( ) E' ( ) D F' ( ) Draw DD', EE' and FF '. E F What relationships exist among these three segments? How do you know? Slide 21 / 154 Writing a Translation Rule Write a translation rule that maps PQRS P'Q'R'S'. P S P' Q S' R Q' R'

Slide 22 / 154 7 In the diagram, ΔA'B'C' is an image of ΔABC. Which rule describes the translation? A T <-5,-3> ( ABC) B T <5,3> ( ABC) C T <-3,-5> ( ABC) T <3,5> ( ABC) D Slide 23 / 154 8 If T <4,-6> (JKLM) = J'K'L'M', what translation maps J'K'L'M' onto JKLM? A T <4,-6> (J'K'L'M') B T <6,-4> (J'K'L'M') C T <6,4> (J'K'L'M') D T <-4,6> (J'K'L'M') Slide 24 / 154 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 coordinates of S' for this translation? A (-6,-4) B (-3,2) C (-3,9) D (-4,13) E none of the above

Slide 25 / 154 Reflections Reflections Activity Lab (Click for link to lab) Return to Table of Contents 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 If a point B is on line m , then the image of B is itself ( B = B'). B B' C If a point C is not on line m , then m is the perpendicular bisector of CC' A' m C' The reflection across m that maps ABC A'B'C' can be written Δ Δ as R m ( ABC) = A'B'C Δ Δ Slide 27 / 154 Reflection When reflecting a figure, reflect the vertices and then draw the sides. Reflect ABCD over line r. Label the vertices of the image. Click here to r see a video D A B C

Slide 28 / 154 Reflection Reflect WXYZ over line s. Label the vertices of the image. W Z X Y s Hint: Turn page so line of symmetry is vertical Slide 29 / 154 Reflection Reflect MNP over line t. Label the vertices of the image. Where is the image of N? Why? M N P t Slide 30 / 154 10 Which point represents the reflection of X? X A point A A B point B B C point C C D point D E None of the above D

Slide 31 / 154 11 Which point represents the reflection of X? A A point A B point B X B C C point C D D point D E none of the above Slide 32 / 154 12 Which point represents the reflection of X? D A point A X B point B C point C A C D point D B E none of the above Slide 33 / 154 13 Which point represents the reflection of D? A point A B point B A D C point C B C D point D E none of these

Slide 34 / 154 14 Is a reflection a rigid motion? Yes No Slide 35 / 154 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. Slide 36 / 154 Reflections in the Coordinate Plane Reflect A, B, & C over the y -axis. How do the coordinates of each point change when the point is reflected over the y -axis? Notation A R y-axis ( A ) = A' R y-axis ( B ) = B' R y-axis ( C ) = C' B C

Slide 37 / 154 Reflections in the Coordinate Plane M Reflect figure JKLM over the x- axis. L K Notation R x-axis ( JKLM) = J'K'L'M' J How do the coordinates of each point change when the point is reflected over the x -axis? Slide 38 / 154 Reflections in the Coordinate Plane Reflect A, B, C & D over the line y = x . B A Notation R y=x ( A ) = A' 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 reflected over the y -axis? Slide 39 / 154 Reflections in the Coordinate Plane A Reflect ABC over x = 2. Δ B C Notation R x =2 ( ABC) = A'B'C' Δ Δ *Hint: draw line of reflection first

Slide 40 / 154 Reflections in the Coordinate Plane Reflect quadrilateral N MNPQ over y = -3 M P Notation R Y=-3 ( MNPQ) = M'N'P'Q' Q Slide 41 / 154 Find the Coordinates of Each Image 1. R x-axis ( A ) 2. R y-axis ( B ) F C A 3. R y=1 ( C ) 4. R x=-1 ( D ) E D B 5. R y=x ( E ) 6. R x=-2 ( F ) Slide 42 / 154 15 The point (4,2) reflected over the x-axis has an image of ______. A (4,2) B (-4,-2) C (-4,2) D (4,-2)

Slide 43 / 154 16 The point (4,2) reflected over the y-axis has an image of _____. A (4,2) B (-4,-2) C (-4,2) D (4,-2) Slide 44 / 154 17 B has coordinates (-3,0). What would be the coordinates of B' if B is reflected over the line x = 1? A (-3,0) B (4,0) C (-3,2) D (5,0) Slide 45 / 154 18 The point (4,2) reflected over the line y=2 has an image of _____. A (4,2) B (4,1) C (2,2) D (4,-2)

Slide 46 / 154 Line of Symmetry A line of symmetry is a line of reflection that divides a figure into 2 congruent halves. These 2 halves reflect onto each other. Slide 47 / 154 Draw Lines of Symmetry Where Applicable C A B D E F Slide 48 / 154 Draw Lines of Symmetry Where Applicable N O M R P Q

Slide 49 / 154 Draw Lines of Symmetry Where Applicable Slide 50 / 154 19 How many lines of symmetry does the following have? A one B two C three D none Slide 51 / 154 20 How many lines of symmetry does the following have? A 10 B 2 C 100 D infinitely many

Slide 52 / 154 21 How many lines of symmetry does the following have? A none B one C nine D infinitely many Slide 53 / 154 22 How many lines of symmetry does the following have? A none J B one C two D infinitely many Slide 54 / 154 23 How many lines of symmetry does the following have? A none H B one C two D infinitely many