Given ABC with A(-1, 1), B(2, 4), and C(4, 1). Translate ABC left 4 units and up 1 unit. a) Vertex matrix: b) Algebraic (arrow) rule:
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1 Unit 7 Transformations 7 Rigid Motion in a Plane Transformation: The operation that maps, or moves, a preimage onto an image. Three basic transformations are reflection, rotation, and translation. Translation Rotation Reflection Image: new figure that results from the transformation of a figure in a plane. Preimage: The original figure in the transformation of a figure in a plane. Isometry: transformation that preserves length, angle measure, parallel lines, distance between points. lso called rigid transformation. Given with (, 1), (, ), and (, 1). Translate left units and up 1 unit. a) Vertex matrix: 1 b) lgebraic (arrow) rule: - - Ex. 1) Using on the front of this sheet, we want to translate right units and down units. Let s call our translated triangle,. a) Draw the graph of on the same set of axes on the front of this sheet. b) What would be the vertex matrix of the original triangle, and the translated triangle?
2 7 Reflections Reflection: transformation that uses a mirror line with an image reflected in the line. Line of Reflection: The mirror line. Line of Symmetry: figure that is mapped onto itself by a reflection in the line. Reflection Theorem: reflection is an isometry. Ex 1: Given with points (, 1), (, ), (, 1), reflect through the x-axis. () Vertex matrices () lgebraic (arrow) rule - Ex : Given with points (, 1), (, ), (, 1), reflect through the y-axis. () Vertex matrices - () lgebraic (arrow) rule
3 Ex : Given with points (, 1), (, ), (, 1), reflect thru the line y=x. () Vertex matrices 1 - () lgebraic (arrow) rule Ex : XYZ with X(, ), Y(, ), Z(, 1). Reflect across the y-axis; write the vertex matrix for the given figure ND its transformed image. Ex : Quadrilateral D with (, 1), (6, ), (, -) and D(1, -6). Reflect across the x-axis; write the vertex matrix for the given figure ND its transformed image.
4 7 Rotations Rotation: type of transformation in which a figure is turned about a fixed point called the center of rotation. Rotation Theorem: rotation is an isometry. Theorem 7.: If likes k and m intersect at point P, then a reflection in k followed by a reflection in m is a rotation about P. The angle of rotation is x, where x is the measure of the acute or right angle formed by k and m. m P" = x k m x P Rotational Symmetry: figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180 degrees or less. Ex 1: Given with points (, 1), (, ), (, 1), rotate 90º clockwise about the origin. () Vertex matrices () lgebraic (arrow) rule 1 -
5 Ex : Given with points (, 1), (, ), (, 1), rotate 180º clockwise about the origin. () Vertex matrices () lgebraic (arrow) rule Ex : Given with points (, 1), (, ), (, 1), rotate 70º clockwise about the origin. () Vertex matrices () lgebraic (arrow) rule - -
6 Review Table Degree lockwise Rotation ounter lockwise Rotation Ex : XYZ with X(, ), Y(, ), Z(, 1). Rotate 90º counterclockwise. Write each vertex matrix. Ex : Quadrilateral D with (, 1), (6, ), (, -) and D(1, -6). Reflect across the x-axis and then rotate the figure 180º. Write each vertex matrix. Ex 6: Pentagon MNOPQ with M(, 0), N(, ), 0(0, 7), P(1, ), and Q(1, 0). Dilate by r = and then rotate the figure 90º clockwise. Write each vertex matrix.
7 7 Translations and Vectors Translation: type of transformation that maps every two points P and Q in the plane to points P and Q, so that the following two properties are true. (1) PP = QQ. () PP QQ or PP &QQ are collinear. Translation Theorem: Translation is an isometry. Theorem 7.: If lines k and m are parallel, then a reflection in line k follow by a reflection in line m is a translation. If P is the image of P, then the following is true: k m 1. PP" is perpendicular to k and m. Q Q Q. PP = d, where d is the distance between k and m. P P d P Vector: quantity that had both direction and magnitude, and is represented by an arrow drawn between two points. omponent form: The form of a vector that combines the horizontal and vertical component of the vector.
8 7- Glide Reflections and ompositions Glide Reflection: transformation in which every point P is mapped onto a point P by the following two steps. (1) translation maps P onto P. () reflection in a line k parallel to the direction of the translation maps P onto P. omposition: The result when two or more transformations are combined to produce a single translation. omposition Theorem: The composition of two (or more) isometries is an isometry.
9 7-6 Frieze Patterns Frieze Pattern: pattern that extends to the left and right is such a way that the pattern can be mapped onto itself by a horizontal translation. K border pattern.
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