Section 12.1 Translations and Rotations Any rigid motion that preserves length or distance is an isometry (meaning equal measure ). In this section, we will investigate two types of isometries: translations and rotations. Translations A translation is a motion of a plane that moves every point of the plane a specified distance in a specified direction along a straight line. Properties of Translations o A figure and its image are congruent. o The image of a line is a line parallel to it. Example: Find the image of AB under the translation from X to X pictured on the dot paper below. B A X X Constructions of Translations!!!" Example: Construct the image A of point A in the direction and magnitude of vector MN. 1
Coordinate Representation of Translations You can use formulas when the translation occurs in the rectangular coordinate system. Property of a Translation in a Coordinate System A translation is a function from the plane to the plane such that to every point (x, y) corresponds the point (x + a, y + b), where a and b are real numbers. Example: Find the coordinates of the image of the vertices of quadrilateral ABCD in the figure below under the translations given. Draw the image in each case. (x, y) (x + 2, y 3) a translation determined by the slide arrow from A(1, 2) to A (3, 1) 2
Rotations A rotation is a transformation of the plane determined by rotating the plane about a fixed point, the center, by a certain amount in a certain direction. Usually, a positive measure is a counterclockwise turn and a negative measure is a clockwise turn. A rotation of 360 about a point will move any point (and figure) onto itself. Such a transformation is an identity transformation. A rotation of 180 is a half-turn. Example: Find the image of ABC under the rotation with center O. A B C Construction of a Rotation Example: Construct the image of point P under a rotation with center O through the angle and in the direction given in the figure below. 3
Slopes of Perpendicular Lines Two lines, neither of which is vertical, are perpendicular iff their slopes m 1 and m 2 satisfy the condition m 1 m 2 = -1. Every vertical line has an undefined slope and is perpendicular to a line with slope 0. Theorem: If y 1 = m 1 x+b 1 and y 2 = m 2 x+b 2 are two distinct non-vertical lines, then a. m 1 = m 2 iff the lines are parallel b. m 1 m 2 = 1 iff the lines are perpendicular. Example: Prove part b of the above theorem. Example: Are the lines 2x 3y = 7 and 3x 2y = 5 parallel, perpendicular, or neither? Example: Find the equation of the line through ( 3, 2) and perpendicular to the line 3x + y = 4. 4
Section 12.2 Reflections and Glide Reflections Reflections A reflection is an isometry in which a figure is reflected across a reflecting line, creating a mirror image. o Unlike a translation or rotation, the reflection reverses the orientation of the original figure. o The reflected figure is congruent to the original figure. Example: Reflect ABC about the line l. Definition of Reflection A reflection in a line l is a transformation of a plane that pairs each point P of the plane with a point P in such a way that l is the perpendicular bisector of P! P, as long as P is not on l. If P is on l, then P = P. Construction of a Reflection Example: Construct the image of point P under a reflection about l. P l 5
Example: Find the line of reflection. Reflections in a Coordinate System Example: Find the image of each ABC under a reflection in line l. a. b. 6
Example: Reflect the point P across the given line and give the coordinates. a. y-axis b. x-axis c. y = x d. y = x Example: Reflect ΔKLM across the line y = x 2. 7
Glide Reflections A glide reflection is another basic isometry. A glide reflection is a transformation consisting of a translation followed by a reflection in a line parallel to the slide arrow. Example: Construct the image of ABC under a glide reflection of slide arrow l. 8
Section 12.3 Size Transformations Definition of Size Transformation A size transformation from the plane to the plane with center O and scale factor r (r > 0) is a transformation that assigns to each point in the plane a point A such that O, A, and A are collinear and OA = r OA and so that O is not between A and A. Theorem 12-1 A size transformation with center O and scale factor r (r > 0) has the following properties: 1. The image of a line segment is a line segment parallel to the original segment and r times as long. 2. The image of an angle is an angle congruent to the original angle. Example: In each figure below, show that ABC is the image of DEF under a size transformation. Identify the center of the size transformation and the scale factor. a. b. 9
Example: Find the image of ABC under a size transformation with center O and scale factor 1 2. Example: Find the image of ABC under a size transformation with center O and scale factor 1.5. 10
Example: Construct the image of quadrilateral ABCD under a size transformation with center O and scale factor 2 3. 11
Definition of Similar Figures Two figures are similar if it is possible to transform one onto the other by a sequence of isometries (translation, rotation, reflection) followed by a size transformation (dilation). Example: Show that ABC is similar to A B C by showing that A B C can be found by performing a sequence of isometries followed by a size transformation to ABC. a. b. 12
Section 12.4 Symmetries Line Symmetries A plane region has a line of symmetry l if it is its own image under a reflection in l. Example: How many lines of symmetry does each object have? Draw the lines of symmetry. M H $ Rotational (Turn) Symmetries A figure has rotational symmetry, or turn symmetry, when the traced figure can be rotated less than 360 about some point P, the turn center, so that it matches the original figure. Example: Find the point P and the rotational symmetry for an equilateral triangle. 13
Example: Determine the amount of the turn for the rotational symmetries of each of the figures below. Z Point Symmetry Any figure that has 180 rotational symmetry is said to have point symmetry about the turn center. Any figure with point symmetry is its own image under a half-turn. This makes the center of the half-turn the midpoint of a segment connecting a point and its image. Example: What objects from previous examples have point symmetry? Example: What kind of triangle has exactly one line of symmetry and no turn symmetries? 14
Symmetries of Three-Dimensional Figures A three-dimensional figure has a plane of symmetry when every point of the figure on one side of the plane has a mirror image on the other side of the plane. Three-dimensional figures with rotational symmetry rotate about an axis of rotation. Example: Determine whether or not each figure has a plane of symmetry. 15
Section 12.5 Tessellations of the Plane A tessellation of a plane (or space) is the filling of the plane (or space) with repetitions of figures in such a way that no figures overlap and there are no gaps. John Locke M.C. Escher Example: Create a tessellation with a square on the grid below: 16
Regular Tesselations A regular tessellation is a tessellation made up of one type of regular polygon. If a regular polygon tessellates the plane, the sum of the congruent angles of the polygons around every vertex must be 360. Example: Try creating tessellations with some basic regular polygons. Which regular polygons tessellate the plane? 17
Semiregular Tessellations When more than one type of regular polygon is used and the arrangement of the polygons at each vertex is the same the tessellation is semiregular. Example: Create a semiregular tessellation consisting only of squares and equilateral triangles. 18
Tessellations with Other Shapes Example: Does every quadrilateral tessellate a plane? Example: Use motion geometry to find other shapes that tessellate a plane. 19