Transformations and the Coordinate Plane Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin

The Coordinates of a Point in a Plane ERHS Math Geometry Mr. Chin-Sung Lin

Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin Two intersecting lines determine a plane. The coordinate plane is determined by a horizontal line, the x-axis, and a vertical line, the y-axis, which are perpendicular and intersect at a point called the origin X Y O

Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin Every point on a plane can be described by two numbers, called the coordinates of the point, usually written as an ordered pair (x, y) X (x, y) Y O

Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin The x-coordinate or the abscissa, is the distance from the point to the y-axis. The y-coordinate or the ordinate is the distance from the point to the x-axis. Point O, the origin, has the coordinates (0, 0) X (x, y) Y O (0, 0) x y

Postulates of Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin Two points are on the same horizontal line if and only if they have the same y-coordinates X (x 2, y) Y O (x 1, y)

Postulates of Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin The length of a horizontal line segment is the absolute value of the difference of the x- coordinates d = |x 2 – x 1 | X (x 2, y) Y O (x 1, y)

Postulates of Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin Two points are on the same vertical line if and only if they have the same x-coordinates X (x, y 2 ) Y O (x, y 1 )

Postulates of Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin The length of a vertical line segment is the absolute value of the difference of the y- coordinates d = |y 2 – y 1 | X (x, y 2 ) Y O (x, y 1 )

Postulates of Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin Each vertical line is perpendicular to each horizontal line X Y O

Locating a Point in the Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin 1.From the origin, move to the right if the x-coordinate is positive or to the left if the x-coordinate is negative. If it is 0, there is no movement 2.From the point on the x-axis, move up if the y- coordinate is positive or down if the y-coordinate is negative. If it is 0, there is no movement X (x, y) Y O x y

Finding the Coordinates of a Point ERHS Math Geometry Mr. Chin-Sung Lin 1.From the point, move along a vertical line to the x- axis.The number on the x-axis is the x-coordinate of the point 2. From the point, move along a horizontal line to the y-axis.The number on the y-axis is the y-coordinate of the point X (x, y) Y O x y

Graphing on the Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin Graph the following points: A(4, 1), B(1, 5), C(-2,1). Then draw ∆ ABC and find its area X Y O

Graphing on the Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin Graph the following points: A(4, 1), B(1, 5), C(-2,1). Then draw ∆ ABC and find its area A (4, 1) C (-2, 1) B (1, 5) D (1, 1) X Y O

Graphing on the Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin Graph the following points: A(4, 1), B(1, 5), C(-2,1). Then draw ∆ ABC and find its area AC = | 4 – (-2) | = 6 BD = | 5 – 1 | = 4 Area = ½ (AC)(BD) = ½ (6)(4) = 12 A (4, 1) C (-2, 1) B (1, 5) D (1, 1) X Y O

Line Reflections ERHS Math Geometry Mr. Chin-Sung Lin

Line Reflections ERHS Math Geometry Mr. Chin-Sung Lin

Line Reflections ERHS Math Geometry Mr. Chin-Sung Lin Y Line of Reflection Line Reflection (Object & Image)

Transformation ERHS Math Geometry Mr. Chin-Sung Lin A one-to-one correspondence between two sets of points, S and S’, such that every point in set S corresponds to one and only one point in set S’, called its image, and every point in S’ is the image of one and only one point in S, called its preimage S S’

A Reflection in Line k ERHS Math Geometry Mr. Chin-Sung Lin 1.If point P is not on k, then the image of P is P’ where k is the perpendicular bisector of PP’ 2.If point P is on k, the image of P is P P’ k P P

Theorem of Line Reflection - Distance ERHS Math Geometry Mr. Chin-Sung Lin Under a line reflection, distance is preserved Given: Under a reflection in line k, the image of A is A’ and the image of B is B’ Prove: AB = A’B’ B’ k B A’ A

Theorem of Line Reflection - Distance ERHS Math Geometry Mr. Chin-Sung Lin Under a line reflection, distance is preserved Given: Under a reflection in line k, the image of A is A’ and the image of B is B’ Prove: AB = A’B’ B’ k B A’ A C D

Theorem of Line Reflection - Distance ERHS Math Geometry Mr. Chin-Sung Lin Under a line reflection, distance is preserved Given: Under a reflection in line k, the image of A is A’ and the image of B is B’ Prove: AB = A’B’ B’ k B A’ A SAS C D

Theorem of Line Reflection - Distance ERHS Math Geometry Mr. Chin-Sung Lin Under a line reflection, distance is preserved Given: Under a reflection in line k, the image of A is A’ and the image of B is B’ Prove: AB = A’B’ B’ k B A’ A CPCTC C D

Theorem of Line Reflection - Distance ERHS Math Geometry Mr. Chin-Sung Lin Under a line reflection, distance is preserved Given: Under a reflection in line k, the image of A is A’ and the image of B is B’ Prove: AB = A’B’ B’ k B A’ A SAS C D

Theorem of Line Reflection - Distance ERHS Math Geometry Mr. Chin-Sung Lin Under a line reflection, distance is preserved Given: Under a reflection in line k, the image of A is A’ and the image of B is B’ Prove: AB = A’B’ B’ k B A’ A CPCTC C D

Theorem of Line Reflection - Distance ERHS Math Geometry Mr. Chin-Sung Lin Since distance is preserved under a line reflection, the image of a triangle is a congruent triangle B’ k B A’ A C C’ M’ M D D’ SSS

Corollaries of Line Reflection ERHS Math Geometry Mr. Chin-Sung Lin Under a line reflection, angle measure is preserved Under a line reflection, collinearity is preserved Under a line reflection, midpoint is preserved B’ k B A’ A C C’ M’ M D D’

Notation of Line Reflection ERHS Math Geometry Mr. Chin-Sung Lin We use r k as a symbol for the image under a reflection in line k r k (A) = A’ r k (∆ ABC ) = ∆ A’B’C’ B’ k B A’ A C C’

Construction of Line Reflection ERHS Math Geometry Mr. Chin-Sung Lin If r k (AC) = A’C’, construct A’C’ k A C

Construction of Line Reflection ERHS Math Geometry Mr. Chin-Sung Lin Construct the perpendicular line from A to k. Let the point of intersection be M k A C M

Construction of Line Reflection ERHS Math Geometry Mr. Chin-Sung Lin Construct the perpendicular line from C to k. Let the point of intersection be N k A C N M

Construction of Line Reflection ERHS Math Geometry Mr. Chin-Sung Lin Construct A’ on AM such that AM = A’M Construct C’ on CN such that CN = C’N k A’ A C C’ N M

Construction of Line Reflection ERHS Math Geometry Mr. Chin-Sung Lin Draw A’C’ k A’ A C C’ N M

Line Symmetry in Nature ERHS Math Geometry Mr. Chin-Sung Lin

Line Symmetry ERHS Math Geometry Mr. Chin-Sung Lin A figure has line symmetry when the figure is its own image under a line reflection This line of reflection is a line of symmetry, or an axis of symmetry

Line Symmetry ERHS Math Geometry Mr. Chin-Sung Lin It is possible for a figure to have more than one axis of symmetry

Line Reflections in the Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin

Reflection in the y-axis ERHS Math Geometry Mr. Chin-Sung Lin Under a reflection in the y-axis, the image of P(a, b) is P’(-a, b) y O x P(a, b) Q(0, b) P’(-a, b)

Reflection in the y-axis ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC is reflected in the y-axis, where A(-3, 3), B(- 4, 1), and C(-1, 1), draw r y-axis (∆ ABC ) = ∆ A’B’C’ y O x B(-4, 1) A(-3, 3) C(-1, 1)

Reflection in the y-axis ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC is reflected in the y-axis, where A(-3, 3), B(- 4, 1), and C(-1, 1), draw r y-axis (∆ ABC ) = ∆ A’B’C’ y O x B’(4, 1) A’(3, 3) C’(1, 1) B(-4, 1) A(-3, 3) C(-1, 1)

Reflection in the x-axis ERHS Math Geometry Mr. Chin-Sung Lin Under a reflection in the x-axis, the image of P(a, b) is P’(a, -b) y O x P(a, b) Q(a, 0) P’(a, -b)

Reflection in the x-axis ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC is reflected in the x-axis, where A(3, 3), B(4, 1), and C(1, 1), draw r x-axis (∆ ABC ) = ∆ A’B’C’ y O x B(4, 1) A(3, 3) C(1, 1)

Reflection in the x-axis ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC is reflected in the x-axis, where A(3, 3), B(4, 1), and C(1, 1), draw r x-axis (∆ ABC ) = ∆ A’B’C’ y O x B(4, 1) A(3, 3) C(1, 1) A’(3, -3) B’(4, -1) C’(1, -1)

Reflection in the Line y = x ERHS Math Geometry Mr. Chin-Sung Lin Under a reflection in the y = x, the image of P(a, b) is P’(b, a) y O x P(a, b) Q(a, a) P’(b, a) R(b, b)

Reflection in the Line y = x ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC is reflected in the x-axis, where A(2, 2), B(1, 4), and C(-1, 1), draw r y=x (∆ ABC ) = ∆ A’B’C’ y O x B(1, 4) A(2, 2) C(-1, 1)

Reflection in the Line y = x ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC is reflected in the x-axis, where A(2, 2), B(1, 4), and C(-1, 1), draw r y=x (∆ ABC ) = ∆ A’B’C’ * Point A is a fixed point since it is on the line of reflection y O x B(1, 4) A(2, 2)=A’(2, 2) C(-1, 1) B’(4, 1) C’(1, -1)

Point Reflections in the Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin

A Point Reflection in P ERHS Math Geometry Mr. Chin-Sung Lin 1.If point A is not point P, then the image of A is A’ and P the midpoint of AA’ 2.The point P is its own image y O x A A’ P

Theorem of Point Reflections ERHS Math Geometry Mr. Chin-Sung Lin Under a point reflection, distance is preserved y O x A P B B’ A’

Theorem of Point Reflections ERHS Math Geometry Mr. Chin-Sung Lin Given: Under a reflection in point P, the image of A is A’ and the image of B is B’ Prove: AB = A’B’ y O x A P B B’ A’

Theorem of Point Reflections ERHS Math Geometry Mr. Chin-Sung Lin Given: Under a reflection in point P, the image of A is A’ and the image of B is B’ Prove: AB = A’B’ y O x A P B B’ A’ SAS

Theorem of Point Reflections ERHS Math Geometry Mr. Chin-Sung Lin Given: Under a reflection in point P, the image of A is A’ and the image of B is B’ Prove: AB = A’B’ y O x A P B B’ A’ CPCTC

Properties of Point Reflections ERHS Math Geometry Mr. Chin-Sung Lin 1.Under a point reflection, angle measure is preserved 2.Under a point reflection, collinearity is preserved 3.Under a point reflection, midpoint is preserved y O x A A’ P

Notation of Point Reflections ERHS Math Geometry Mr. Chin-Sung Lin We use R p as a symbol for the image under a reflection in point P R p (A) = B means “The image of A under a reflection in point P is B.” R (1,2) (A) = A’means “The image of A under a reflection in point (1, 2) is A’.”

Point Symmetry ERHS Math Geometry Mr. Chin-Sung Lin A figure has point symmetry if the figure is its own image under a reflection in a point

Point Symmetry ERHS Math Geometry Mr. Chin-Sung Lin Other examples of figures that have point symmetry are letters such as S and N and numbers such as 8 S N 8

Reflection in the Origin ERHS Math Geometry Mr. Chin-Sung Lin Under a reflection in the origin, the image of P(a, b) is P’(-a, -b) R O (a, b) = (-a, -b) y O x P(a, b) P’(-a, -b)

Reflection in the Origin ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC is reflected in the origin, where A(-3, 3), B(-4, 1), and C(-1, 1), draw R O (∆ ABC ) = ∆ A’B’C’ y O x B(-4, 1) A(-3, 3) C(-1, 1)

Reflection in the Origin ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC is reflected in the origin, where A(-3, 3), B(-4, 1), and C(-1, 1), draw R O (∆ ABC ) = ∆ A’B’C’ y O x B’(4, -1) A’(3, -3) C’(1, -1) B(-4, 1) A(-3, 3) C(-1, 1)

Reflection in the point ERHS Math Geometry Mr. Chin-Sung Lin (A)What are the coordinates of B, the image of A(-3, 2) under a reflection in the origin? (B) What are the coordinates of C, the image of A(-3, 2) under a reflection in the x-axis? (C)What are the coordinates of D, the image of C under a reflection in the y-axis? (D)Does a reflection in the origin give the same result as a reflection in the x-axis followed by a reflection in the y-axis? Justify your answer.

Translations in the Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin

Translation ERHS Math Geometry Mr. Chin-Sung Lin A translation is a transformation of the plane that moves every point in the plane the same distance in the same direction y O x B’ B C’ C A’ A

Translation ERHS Math Geometry Mr. Chin-Sung Lin In the coordinate plane, the distance is given in terms of horizontal distance (change in the x-coordinates) and vertical distance (change in the y-coordinates) y O x B’ B C’ C A’ A x-coor. y-coor.

Translation ERHS Math Geometry Mr. Chin-Sung Lin A translation of a units in the horizontal direction and b units in the vertical direction is a transformation of the plane such that the image of P(x, y) is P’(x + a, y + b) y x P’(x + a, y + b) P(x, y) a b

Translation ERHS Math Geometry Mr. Chin-Sung Lin The image of P(x, y) is P’(x + a, y + b), if the translation moves a point to the right, a > 0 if the translation moves a point to the left, a < 0 if the translation moves a point up, b > 0 if the translation moves a point down, b < 0 y x P’(x + a, y + b) P(x, y) a b

Theorem of Translation ERHS Math Geometry Mr. Chin-Sung Lin Under a translation, distance is preserved y O x B A B’ A’

Theorem of Translation ERHS Math Geometry Mr. Chin-Sung Lin Given: A translation in which the image of A(x 1,y 1 ) is A’(x 1 +a, y 1 +b) and the image of B(x 2, y 2 ) is B’(x 2 +a, y 2 +b) Prove: AB = A’B’ y O x A’ (x 1 +a, y 1 +b) B (x 2, y 2 ) A (x 1, y 1 ) B’ (x 2 +a, y 2 +b)

Theorem of Translation ERHS Math Geometry Mr. Chin-Sung Lin Given: A translation in which the image of A(x 1,y 1 ) is A’(x 1 +a, y 1 +b) and the image of B(x 2, y 2 ) is B’(x 2 +a, y 2 +b) Prove: AB = A’B’ y O x B (x 2, y 2 ) A (x 1, y 1 ) B’ (x 2 +a, y 2 +b) A’ (x 1 +a, y 1 +b)

Theorem of Translation ERHS Math Geometry Mr. Chin-Sung Lin Given: A translation in which the image of A(x 1,y 1 ) is A’(x 1 +a, y 1 +b) and the image of B(x 2, y 2 ) is B’(x 2 +a, y 2 +b) Prove: AB = A’B’ y O x B’ (x 2 +a, y 2 +b) A’ (x 1 +a, y 1 +b) |x 1 -x 2 | |y 1 -y 2 | B (x 2, y 2 ) A (x 1, y 1 )

Theorem of Translation ERHS Math Geometry Mr. Chin-Sung Lin Given: A translation in which the image of A(x 1,y 1 ) is A’(x 1 +a, y 1 +b) and the image of B(x 2, y 2 ) is B’(x 2 +a, y 2 +b) Prove: AB = A’B’ y O x B (x 2, y 2 ) A (x 1, y 1 ) B’ (x 2 +a, y 2 +b) A’ (x 1 +a, y 1 +b) |x 1 -x 2 | |y 1 -y 2 | SAS & CPCTC

Properties of Translation ERHS Math Geometry Mr. Chin-Sung Lin 1.Under a translation, angle measure is preserved 2.Under a translation, collinearity is preserved 3.Under a translation, midpoint is preserved y O x A’ (x 1 +a, y 1 +b) B (x 2, y 2 ) A (x 1, y 1 ) B’ (x 2 +a, y 2 +b)

Notation of Translation ERHS Math Geometry Mr. Chin-Sung Lin We use T a, b as a symbol for the image under a translation of a units in the horizontal direction and b units in the vertical direction T a, b (x, y) = (x + a, y + b)

Translation ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under T 7,1 y O x B(-4, 1) A(-3, 3) C(-1, 1)

Translation ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under T 7,1 y O x B(-4, 1) A(-3, 3) C(-1, 1) B’(3, 2) A’(4, 4) C’(6, 2)

Translational Symmetry ERHS Math Geometry Mr. Chin-Sung Lin A figure has translational symmetry if the image of every point of the figure is a point of the figure

Translational Symmetry ERHS Math Geometry Mr. Chin-Sung Lin

Rotations in the Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin

Rotation ERHS Math Geometry Mr. Chin-Sung Lin A rotation is a transformation of a plane about a fixed point P through an angle of d degrees such that: 1.For A, a point that is not the fixed point P, if the image of A is A’, then PA = PA’ and m  APA’ = d 2.The image of the center of rotation P is P y O x P A’ A d

Theorem of Rotation ERHS Math Geometry Mr. Chin-Sung Lin Distance is preserved under a rotation about a fixed point P A’ A B’ B

Theorem of Rotation ERHS Math Geometry Mr. Chin-Sung Lin Given: P is the center of rotation. If A is rotated about P to A’, and B is rotated the same number of degrees to B’ Prove: AB = A’B’ P A’ A B’ B d d

Theorem of Rotation ERHS Math Geometry Mr. Chin-Sung Lin Given: P is the center of rotation. If A is rotated about P to A’, and B is rotated the same number of degrees to B’ Prove: AB = A’B’ P A’ A B’ B m  APA’ = m  BPB’ m  APB = m  A’PB’

Theorem of Rotation ERHS Math Geometry Mr. Chin-Sung Lin Given: P is the center of rotation. If A is rotated about P to A’, and B is rotated the same number of degrees to B’ Prove: AB = A’B’ P A’ A B’ B SAS

Theorem of Rotation ERHS Math Geometry Mr. Chin-Sung Lin Given: P is the center of rotation. If A is rotated about P to A’, and B is rotated the same number of degrees to B’ Prove: AB = A’B’ P A’ A B’ B CPCTC

Properties of Rotation ERHS Math Geometry Mr. Chin-Sung Lin 1.Under a rotation, angle measure is preserved 2.Under a rotation, collinearity is preserved 3.Under a rotation, midpoint is preserved P A’ A B’ B

Notation of Rotation ERHS Math Geometry Mr. Chin-Sung Lin We use R P, d as a symbol for the image under a rotation of d degrees about point P A rotation in the counterclockwise direction is called a positive rotation A rotation in the clockwise direction is called a negative rotation R O, 30 o (A) = Bthe image of A under a rotation of 30° degrees about the origin is B

Notation of Rotation ERHS Math Geometry Mr. Chin-Sung Lin The symbol R is used to designate both a point reflection and a rotation 1.When the symbol R is followed by a letter that designates a point, it represents a reflection in that point (e.g., R P ) 2.When the symbol R is followed by both a letter that designates a point and the number of degrees, it represents a rotation of the given number of degrees about the given point (e.g., R O, 30 o ) 3.When the symbol R is followed by the number of degrees, it represents a rotation of the given number of degrees about the origin (e.g., R 90 o )

Theorem of Rotation ERHS Math Geometry Mr. Chin-Sung Lin Under a counterclockwise rotation of 90° about the origin, the image of P(a, b) is P’(–b, a) R O,90° (x, y) = (-y, x) orR 90° (x, y) = (-y, x)

Rotation ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under R O,90 o y O x B(-4, 1) A(-3, 3) C(-1, 1)

Rotation ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under R O,90 o y O x B(-4, 1) A(-3, 3) C(-1, 1) A’(-3, -3) B’(-1, -4) C’(-1, -1)

Rotation ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A”B”C”, the image of ∆ ABC under R O,180 o y x B(-4, 1) A(-3, 3) C(-1, 1)

Rotation ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A”B”C”, the image of ∆ ABC under R O,180 o y O x B(-4, 1) A(-3, 3) C(-1, 1) A’(-3, -3) B’(-1, -4) C’(-1, -1) C”(1, -1) A”(3, -3) B”(4, -1)

Rotation ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A”B”C”, the image of ∆ ABC under R O,180 o y O x B(-4, 1) A(-3, 3) C(-1, 1) C”(1, -1) A”(3, -3) B”(4, -1)

Rotation 180 o = Point Reflection ERHS Math Geometry Mr. Chin-Sung Lin ∆ A”B”C”, the image of ∆ ABC under R O,180 o is the same as the image of ∆ ABC under point reflection R O y O x B(-4, 1) A(-3, 3) C(-1, 1) C”(1, -1) A”(3, -3) B”(4, -1)

Rotational Symmetry ERHS Math Geometry Mr. Chin-Sung Lin A figure is said to have rotational symmetry if the figure is its own image under a rotation and the center of rotation is the only fixed point

Rotational Symmetry ERHS Math Geometry Mr. Chin-Sung Lin Many letters, as well as designs in the shapes of wheels, stars, and polygons, have rotational symmetry S H 8 Z N

Glide Reflections in the Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin

Composition of Transformations ERHS Math Geometry Mr. Chin-Sung Lin When two transformations are performed, one following the other, we have a composition of transformations

Glide Reflection ERHS Math Geometry Mr. Chin-Sung Lin A glide reflection is a composition of transformations of the plane that consists of a line reflection and a translation in the direction of the line of reflection performed in either order y x B B’ A A’ C’ C B” A” C”

Glide Reflection ERHS Math Geometry Mr. Chin-Sung Lin A glide reflection is a composition of transformations of the plane that consists of a line reflection and a translation in the direction of the line of reflection performed in either order y x B B’ A A’ C’ C B” A” C”

Theorem of Glide Reflection ERHS Math Geometry Mr. Chin-Sung Lin Under a glide reflection, distance is preserved y x B B’ A A’ C’ C B” A” C”

Properties of Glide Reflection ERHS Math Geometry Mr. Chin-Sung Lin 1.Under a glide reflection, angle measure is preserved 2.Under a glide reflection, collinearity is preserved 3.Under a glide reflection, midpoint is preserved y x B B’ A A’ C’ C B” A” C”

Isometry ERHS Math Geometry Mr. Chin-Sung Lin An isometry is a transformation that preserves distance All five transformations: 1.line reflection, 2.point reflection, 3.translation, 4.rotation, and 5.glide reflection. Each of these transformations is called an isometry

Glide Reflection ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under r y-axis, and ∆ A”B”C”, the image of ∆ A’B’C’ under T 0, –4 y O x B(-4, 1) A(-3, 3) C(-1, 1)

Glide Reflection ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under r y-axis, and ∆ A”B”C”, the image of ∆ A’B’C’ under T 0, –4 y O x B(-4, 1) A(-3, 3) C(-1, 1) A’(3, 3) B’(4, 1) C’(1, 1)

Glide Reflection ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC has vertices A(-3, 3), B(-4, 1), and C(-1, 1), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under r y-axis, and ∆ A”B”C”, the image of ∆ A’B’C’ under T 0, –4 y O x B(-4, 1) A(-3, 3) C(-1, 1) A’(3, 3) B’(4, 1) C’(1, 1) A”(3, –1) B”(4, –3) C”(1, – 3)

Glide Reflection ERHS Math Geometry Mr. Chin-Sung Lin The vertices of ∆PQR are P(2, 1), Q(4, 1), and R(4, 3) 1.Find ∆P’Q’R’, the image of ∆PQR under r y=x followed by T –3, –3 2.Find ∆P”Q”R”, the image of ∆PQR under T –3, –3 followed by r y=x 3.Are ∆P’Q’R’ and ∆P”Q”R” the same triangle? 4.Are r y=x followed by T –3, –3 and T –3, –3 followed by r y=x the same glide reflection? Explain 5.Write a rule for this glide reflection

Dilations in the Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin

Dilation ERHS Math Geometry Mr. Chin-Sung Lin A dilation of k is a transformation of the plane such that: 1.The image of point O, the center of dilation, is O 2.When k is positive and the image of P is P’, then OP and OP’ are the same ray and OP’ = kOP 3.When k is negative and the image of P is P’, then OP and OP’ are opposite rays and OP’ = -kOP. y O x P P’ k > 0 P’ k < 0

Notation of Dilations ERHS Math Geometry Mr. Chin-Sung Lin We use D k as a symbol for the image under a dilation of k with center at the origin P (x, y)  P’ (kx, ky) orD k (x, y) = (kx, ky) D 2 (3, 4) = (6, 8)

Dilation ERHS Math Geometry Mr. Chin-Sung Lin Under a dilation about a fix point, distance is not preserved, and angle measurement is preserved Dilation is not an isometry y O x A A’ B B’

Glide Reflection ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC has vertices A(2, 1), B(1, 3), and C(3, 2), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under D 2 y O x B(1, 3) A(2, 1) C(3, 2)

Glide Reflection ERHS Math Geometry Mr. Chin-Sung Lin If ∆ ABC has vertices A(2, 1), B(1, 3), and C(3, 2), find the coordinates of the vertices of ∆ A’B’C’, the image of ∆ ABC under D 2 y O x B(1, 3) A(2, 1) C(3, 2) B’(2, 6) A’(4, 2) C’(6, 4)

Transformations as Functions ERHS Math Geometry Mr. Chin-Sung Lin

Functions ERHS Math Geometry Mr. Chin-Sung Lin A function is a set of ordered pairs in which no two pairs have the same first element The set of first elements is the domain of the function and the set of second elements is the range Domain Range

Transformations as Functions ERHS Math Geometry Mr. Chin-Sung Lin Transformation can be viewed as a one-to-one function S S’

Notations of Functions ERHS Math Geometry Mr. Chin-Sung Lin For example, y = x + 1 is a function f, it can represented as: y = x + 1 f(x) = x + 1 f: x -> x + 1 f = { (x, y) | y = x + 1} y and f(x) both represent the second element of the ordered pair

Composition of Transformations ERHS Math Geometry Mr. Chin-Sung Lin When two transformations are performed, one (f) following the other (g), we have a composition of transformations y = g( f(x) ) or y = g o f

Composition of Transformations ERHS Math Geometry Mr. Chin-Sung Lin A’ is the image of A(2, 5) under a reflection in the line y = x followed by the translation T 2,0, we can write T 2, 0 (r y = x (A)) = A’orT 2, 0 o r y = x (A) = A’ A’ = T 2, 0 (r y = x (2, 5)) = T 2, 0 o r y = x (2, 5) = T 2, 0 (5, 2) = (7, 2)

Orientation ERHS Math Geometry Mr. Chin-Sung Lin In a figure, the vertices, when traced from A to B to C to …. are in the clockwise or the counter-clockwise direction, called the orientation of the points A C B Clockwise Orientation

Direct Isometry ERHS Math Geometry Mr. Chin-Sung Lin A direct isometry is a transformation that preserves distance and orientation The following three transformations: 1.point reflection, 2.translation, and 3.rotation each of these transformations is direct isometry

Opposite Isometry ERHS Math Geometry Mr. Chin-Sung Lin An opposite isometry is a transformation that preserves distance, but changes the orientation The following two transformations: 1.line reflection, and 2.glide reflection each of these transformations is opposite isometry

Q & A ERHS Math Geometry Mr. Chin-Sung Lin

The End ERHS Math Geometry Mr. Chin-Sung Lin