Section 12.1 Translations and Rotations Any rigid motion that preserves length or distance is an isometry. We look at two types of isometries in this section: translations and rotations. Translations A translation is a motion of a plane that moves every point of the plane in a specified distance in a specified direction along a straight line (which can be shown by a slide-arrow or vector). Example 1: Find the image of AB under the translation from X to X pictured on the dot paper below. Properties of Translations A figure and its image are congruent. The image of a line is a line parallel to it. Constructions of Translations To construct the image A of point A in the direction and magnitude of vector MN, construct a parallelogram MAA N so that AA is in the same direction as MN Example 2: Given point A and vector MN, construct the image, A, of A. A M N 1
Coordinate Representation of Translations Formulas can be used when the translation is done in the rectangular coordinate system. Example 3: Let s look at the following translation in a 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 4: Find the coordinates of the image of the vertices of quadrilateral ABCD under each of the following translations: a) (x,y) (x 2,y+4) b) A translation determined by the slide arrow from (4, 3) to (2,1). 2
Rotation 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. Example 5: Find the image of ABC under the rotation with center O. Construction of a Rotation To construct the image of point P under a rotation with center O through a given angle A in the direction indicated: Construct an isosceles triangle BAC with B on one side of the given angle and C on the other side so that AB=AC=OP. Construct POP congruent to BAC. Example 6: Construct the image of point P under the rotation with center O through the angle and in the direction given below: P A O 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 about a point is a half-turn. 3
Slopes of Perpendicular Lines 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 7: Prove part(b) of the above theorem. 4
Example 8: Are the lines 2x 3y = 7 and 3x 2y = 5 parallel, perpendicular, or neither? Example 9: Find the equation of the line through ( 3,2) and perpendicular to the line 3x+y = 4. Section 12.1 Homework Problems: 1-8, 11, 13, 18, 19, 22, 33, 34 5
Section 12.2 - Reflections and Glide Reflections Reflections A reflection is an isometry in which the figure is reflected across a reflecting line, creating a mirror image. Unlike a translation or rotation, the reflection reverses the orientation of the original figure but the reflected figure is still congruent to the original figure. Example 1: Reflect ABC about the line l. B C l A Definiton 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 PP, as long as P is not on l. If P is on l, then P = P. Example 2: Construct the image of point P under a reflection about l. P l 6
Example 3: Find the line of reflection of the point P and its image, P. P P Example 4: Construct the image of AB under a reflection in line m. m B A 7
Constructing a Reflection on Dot Paper Example 5: Find the image of each ABC under a reflection in line l. A B C a) l C b) l A B Reflections in a Coordinate System Example 6: Reflect the point P across the indicated line and give the coordinates. a) x-axis 8
b) y-axis c) y = x 9
d) y = x Example 7: Reflect KLM across the line y = x 2 10
Glide Reflections: A glide reflection is another basic isometry. It is a transformation consisting of a translation followed by a reflection in a line parallel to the slide arrow. Example 8: Construct the image of ABC under a glide reflection of slide arrow l. B A C l Section 12.2 Homework Problems: 1, 4, 5, 9-11, 13, 14, 35-38 11
Definition of Size Transformation Section 12.3 - Size Transformations 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 A in the plane a point A, such that O, A, and A are collinear and OA = r OA and so that OA is not between A and A. A A C B C B O Example 1: Find the image of ABC under the size transformation with center O and scale factor 1 2 A C O B 12
Example 2: Find the image of ABC under the size transformation with center O and scale factor 1.5 A C O B Example 3: Construct the image of point P under a size transformation with center O and scale factor 1 2 P O 13
Example 4: Construct the image of point P under a size transformation with center O and scale factor 2 P O Example 5: Construct the image of point P under a size transformation with center O and scale factor 2 3 P O 14
Example 6: Construct the image of quadrilateral ABCD under a size transformation with center O and scale factor 2 3 B A C D O 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. Definition of Similar Figures: Two figures are similar if it is possible to transform one onto the other by a sequence of isometries followed by a size transformation. 15
Example 7: 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. C C A B B A Example 8: 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. C B C A A B Section 12.3 Homework Problems: 1-4, 6-8, 13, 15, 21, 22 16
Section 12.4 - Symmetries Line Symmetries A plane region has a line l of symmetry if a reflection of the plane about l produces exactly the same figure. Example 1: How many lines of symmetry does each object have? Draw the lines of symmetry. 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 2: Find the point P and the rotational symmetry for an equilateral triangle. 17
Example 3: Determine the amount of turn for the rotational symmetries for each of the figures shown below, if they exist. 18
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 4: Determine whether or not the following figures have point symmetry. 19
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. Example 5: Determine whether or not each figure has a plane of symmetry. s h s s h l r b 20
Example 6: Describe what kind of triangle has exactly one line of symmetry and no turn symmetries. Example 7: Given an arc of a circle, find its center and radius. Section 12.4 Homework Problems: 2-10, 12, 13b 21
Section 12.5 - Tessellations of the Plane A tessellation of a plane is the filling of the plane with repetitions of figures in such a way that no figures overlap and there are no gaps. Example 1: Create a tessellation with a square on the grid below: 22
Regular Tessellations A regular tesselation is a tessellation made up of one type of regular polygon. Example 2: Try creating tessellations with some basic regular polygons. Which regular polygons tessellate the plane? 23
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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 3: Create a semiregular tessellation consisting only of squares and equilateral triangles. Tessellating with Other Shapes There has been a lot of thought put into determining what shapes (other than regular polygons) tessellate a plane. Example 4: Does any quadrilateral tessellate a plane? 25
Example 5: Let s see how motion geometry can be used to find other shapes that tessellate a plane. Section 12.5 Homework Problems: 1-6, 8, 12, 13 26