5.7 Rotations
English5.7SpanishRotationsWhat are the three basic ways to move anSSTATESTANDARDSobject in a plane?MA.7.G.4.2RotateA bicycle wheel1112can rotate clockwise1112108716258623984711039412111103912or counterclockwise.47556ACTIVITY: Three Basic Ways to Move ThingsThere are three basic ways to move objects on a flat surface.1. Translate the object.2. Reflect the object.3. Rotate the object.Work with a partner. 232Chapter 5yCut out a paper triangle that is thesame size as the blue triangle shown.543Decide how you can move the bluetriangle to make each red triangle. Is each move a translation, a reflection,or a rotation? Draw four other red triangles in acoordinate plane. Describe how youcan move the blue triangle to makeeach red triangle.Similarity and Transformations21 3 2O 2 3 4 612345678 x
EnglishSpanish2ACTIVITY: Tessellating a PlaneWork with a partner.a. Describe how the figure labeled 1 in each diagram can be moved to makethe other figures.Triangles1Quadrilaterals1b. EXPERIMENT Will any triangle tessellate? Conduct an experiment to gatherinformation to help form your conclusion. Draw a triangle. Cut it out. Thenuse it to trace other triangles so that you cover the plane with triangles thatare all the same shape.c. EXPERIMENT Will any quadrilateral tessellate? Conduct an experiment togather information to help form your conclusion. Draw a quadrilateral. Cut itout. Then use it to trace other quadrilaterals so that you cover the plane withquadrilaterals that are all the same shape.3. IN YOUR OWN WORDS What are the three basic ways to move an objectin a plane? Draw an example of each.“Dear Sub Shop: Why do you put thecheese on the subs so some parts havedouble coverage and some have none?”“My suggestion is that you use thetessellation property of triangles for evencheese coverage.”Use what you learned about rotations to complete Exercises 7–9on page 236.Section 5.7MSFL7PE 0507.indd 233Rotations23310/20/09 3:25:06 PM
English5.7SpanishLessonLesson TutorialsKey Vocabularyrotation, p. 234center of rotation,p. 234angle of rotation,p. 234RotationsA rotation, or turn, is atransformation in which a figureis rotated about a point called thecenter of rotation. The number ofdegrees a figure rotates is the angleof rotation.turnangle of rotationThe original figure and its imagehave the same size and shape.center of rotationEXAMPLE1Standardized Test PracticeYou must rotate the puzzle piece 270 clockwiseabout point P to fit it into a puzzle. Which piecefits in the puzzle as shown?A B C PD Rotate the puzzle piece 270 clockwise about point P.Study TipWhen rotating figures,it may help to sketchthe rotation in severalsteps, as shown inExample 1.turn 270 PThe correct answer is C .1. Which piece is a 90 counterclockwise rotation about point P ?Exercises 7–12234Chapter 52. Is choice D a rotation of the original puzzle piece? If not, whatkind of transformation does the image show?Similarity and Transformations
EnglishSpanishEXAMPLE2Rotating a FigureThe vertices of a trapezoid are W( 4, 2), X( 3, 4), Y( 1, 4), andZ( 1, 2). Rotate the trapezoid 180 clockwise about the origin. Whatare the coordinates of the image?XDraw WXYZ.Yy43W 4 3 2Z2turn 180 O 2 3 4234 xW′Plot Z′, W′, X′, and Y′.Connect the vertices.Z′Y′X′The coordinates of the image are W ′(4, 2), X ′(3, 4), Y ′(1, 4),and Z′(1, 2).EXAMPLE3Rotating a FigureThe vertices of a triangle are J(1, 2), K(4, 2), and L(1, 3). Rotate thetriangle 90 counterclockwise about vertex L. What are the coordinatesof the image?Draw JKL.yCommon Error43Be sure to payattention to whether arotation is clockwise orcounterclockwise.2JK1K′ 2O2Plot J′ and K′.Connect the vertices.34 xturn 90 J′ 4LThe coordinates of the image are J′( 4, –3), K ′( 4, 0), and L′(1, 3).3. A triangle has vertices Q(4, 5), R(4, 0), and S(1, 0).Exercises 13–16a. Rotate the triangle 90 counterclockwise about the origin.b. Rotate the triangle 180 about vertex S.c. Are the images in parts (a) and (b) the same size andshape? Explain.Section 5.7Rotations235
EnglishSpanishExercises5.7Help with Homework1. VOCABULARY Identify the transformation shown.a.b.c.2. VOCABULARY What are the coordinates of the center of rotation in Example 2?Example 3?MENTAL MATH A figure lies entirely in Quadrant II. In which quadrant will thefigure lie after the given clockwise rotation about the origin?3. 90 4. 180 5. 270 6. 360 6) 39 (- 3) 3 (- 9) 4 (- 1)9 (-Tell whether the blue figure is a rotation of the red figure about the origin.If so, give the angle and direction of rotation.17.8.y49.yy4332 4 3 2O123 4 3 24 x2211O123 4 3 24 xO23 2 2 2 3 3 3 4 4 41A figure has rotational symmetry if a rotation of 180 or less produces an image that fitsnal symmetry.exactly on the original figure. Explain why the figure has rotational10.236Chapter 511.Similarity and Transformations12.4 x
EnglishSpanishThe vertices of a parallelogram are A( 4, 1), B( 3, 4), C( 1, 4), and D( 2, 1). Rotate theparallelogram as described. Find the coordinates of the image.2313. 90 counterclockwise about the origin14. 270 clockwise about the origin15. 180 clockwise about vertex D16. 90 counterclockwise about vertex B17. WRITING Why is it not necessary to use the words clockwiseand counterclockwise when describing a rotation of 180 ?18. DILATIONS A dilation is a transformation in which a figureis enlarged or reduced.y876a. Dilate Rectangle JKLM by multiplying the x- andy-coordinates of each vertex by 2. Compare theoriginal figure and its image.5J4M3b. Are the rectangles identical? Are they similar? Explain.c. How do dilations differ from translations, reflections,and rotations?21KOL1234567 xyx20.19. TREASURE MAP You want to findthe treasure located on the mapat . You are located at . Thefollowing transformations will leadyou to the treasure, but they arenot in the correct order. Find thecorrect order. Use eachtransformation exactly once. Rotate 180 about the origin. Reflect in the y-axis. Rotate 90 counterclockwise aboutthe origin. Translate 1 unit right and 1 unit up.A triangle is rotated 90 counterclockwise about the origin. Its imageis translated 1 unit left and 2 units down. The vertices of the final triangle are( 5, 0), ( 2, 2), and ( 2, 1). What are the vertices of the original triangle?Identify the solid. . MULTIPLE CHOICE What is the value of x y when x 5 and y 8?(Section 1.3)A 13 B 3 C 3 D 13 Section 5.7MSFL7PE 0507.indd 237SECTION 1.3Rotations23710/20/09 3:25:28 PM
rotation, p. 234 center of rotation, p. 234 angle of rotation, p. 234 Rotations A rotation, or turn, is a turn angle of rotation center of rotation transformation in which a fi gure is rotated about a point called the center of rotation. The number of degrees a fi gure rotates is the angle of rotation
rotations are performed. Indeed, Eq. (1) is a linear transformation that can be represented by a matrix, and matrices in general do not commute. However, for small rotations at linear order in δχthe result does not depend on the order of rotations. As an example consider two successive rotations r(1) r h δχ(1) r i, r(2) r(1) h .
Rotations averaging is the problem of assigning a rotation matrix to every vertex in a graph, in a way that best respects given relative rotations on each edge. This problem has become a staple of recent global Structure from Motion (SfM) methods, where the vertices represent camera
Synchronization of rotations is the problem of estimating rotation matrices R 1;:::;R N from noisy measurements of relative rotations R i R j. The set of available measurements gives rise to a graph structure, where the Nnodes correspond to the rotations R i and an edge is present between two
Using Rotations to Obtain Compatibility Equations for Statically Indeterminate Trusses Harry E. Williams1 and Clive L. Dym, P.E., F.ASCE2 Abstract: This paper presents a new way of determining the compatibility conditions for statically indeterminate trusses. It is based on equations relating the angular displacements (rotations) of truss elements to the corresponding extensions obtained by .
focus on the optical observational data, and we study the statistical properties of the detected EVPA rotations in both observing sea-sons. We aim to determine the average parameters of the rotations, and test possible correlations between these parameters as well as the average total flux density and fractional polarization. The in-
Class- VI-CBSE-Mathematics Knowing Our Numbers Practice more on Knowing Our Numbers Page - 4 www.embibe.com Total tickets sold ̅ ̅ ̅̅̅7̅̅,707̅̅̅̅̅ ̅ Therefore, 7,707 tickets were sold on all the four days. 2. Shekhar is a famous cricket player. He has so far scored 6980 runs in test matches.
1 Rotations and angular momentum Rotation about one or more axes is a common and useful symmetry of many physical systems. Central . In atomic physics, we also deal with . but the length we are concerned with in quantum mechanis is the hermitian normofastate: h j i Letarotationofthestatej ibedenotedby j i D (n;’)j i 6.
(PY2). The Advanced Pharmacy Practice Experience rotations will be offered in the Fall, Winter, Spring Quarters of the fourth professional year (third calendar year). The experiential rotations are designed to provide students the opportunity to focus on the traditional and the clinical aspects of pharmacy
