Topic 3: Goals and Common Core Standards Ms. Helgeson Identify the three basic rigid transformations Identify and use reflections in a plane Identify relationships between reflections and line symmetry. Identify rotations in a plane Identify and use translations in a plane Use vectors in real-life situations.

Identify glide reflections in a plane. Represent transformations as compositions of simpler transformations. CC.9-12.G.CO.5 CC.9-12.G.CO.4 CC.9-12.G.CO.2 CC.9-12.G.CO.3

Topic 3 Transformations

Symmetry—The Balance We See in Nature and Culture Symmetry in Nature – we find an endless display of geometric patterns and shapes. One of the most common characteristics in nature (and art) is symmetry. Symmetry makes us think of balance, harmony, sameness on opposite sides.

Broccoli is just one of the many instances of fractal symmetry in nature. a fractal is a complex pattern where each part of a thing has the same geometric pattern as the whole. So with broccoli, each floret presents the same logarithmic spiral as the whole head. the entire veggie is one big spiral composed of smaller, cone-like buds that are also mini-spirals. The honeycomb is a case of wallpaper symmetry, where a repeated pattern covers a plane.

Sunflowers boast radial symmetry and an interesting type of numerical symmetry known as the Fibonacci sequence. The Fibonacci sequence is 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. If we took the time to count the number of seed spirals in a sunflower, we’d find that the amount of spirals adds up to a Fibonacci number. Spider web Peacock snowflake

Golden rectangle

3.1 and part of 3.5 Reflections and Symmetry Goal: Identifying Transformations Figures in a plane can be reflected, rotated, or translated to produce new figures.

1. Pre-image (input) - Original figure Image (output) - New figure 4. Transformation-The operation that maps, or moves, the pre-image onto the image.

For example, the transformation T moves point A to A´ For example, the transformation T moves point A to A´. A transformation is sometimes called a mapping. Transformation T maps point A to point A´. A´ A T Image Pre-image

Three basic rigid transformations are reflections, rotations, and translations.

B' C' A' A C B This is a reflection in the x-axis.

A B C' C B' A' This is a rotation about the origin.

B' B C' A' A C The transformation is a translation. The image was obtained by sliding ∆ABC up and to the right.

6. Isometry-Transformation that preserves lengths 6. Isometry-Transformation that preserves lengths. Isometries also preserve angle measures, parallel lines, and distances between points. Transformations that are isometries are called rigid transformations.

Does the transformation appear to be an isometry?

You can also use arrow notation as follows: ∆ABC →∆DEF (∆ABC is mapped onto ∆DEF) The order in which the vertices are listed specifies the correspondence. A → D B → E C → F B E → corresponds F A C D

Reflections Reflection: Reflection is a rigid motion, meaning an object changes its position but not its size or shape. In a reflection, you create a mirror image of the object. There is a particular line (line of symmetry) that acts like the mirror.

Theorem: Reflection Theorem A reflection is an isometry.

Reflections in the coordinate axes have the following properties: a) If (x, y) is reflected in the x-axis, its image is the point (x, -y). b) If (x, y) is reflected in the y-axis, its image is the point (-x, y). c) If (x, y) is reflected in the line y = x, its image is the point (y, x) d) If (x, y) is reflected in the line y = -x, its image is the point (-y, -x)

Graph the given reflection. J(2, 6) in the y-axis. J (2, 6) = ? K M Graph the given reflection. J(2, 6) in the y-axis. J (2, 6) = ? K(7, 3) in the x-axis. K (7, 3) = ? M(-3, -2) in the line x = 2. M (-3, -2) = ? J(2, 6) in the line y = x. J (2, 6) = ? M(-3, -2) in the line y = -x M (-3, -2) = ? Y-axis X-axis X = 2 Y = x Y = -x

Finding a Minimum Distance Two houses A(1, 5) and B(7, 1) are located on a rural road m. You want to place a telephone pole on the road at point C so that the length of the telephone cable, AC + BC, is a minimum. Where should you locate C?

m m Reflectional Symmetry A line of symmetry for a plane figure is a line in which, when the figure is reflected, its image is itself. m m The reflection R map the figure onto itself. Observe that line m divide the figure into two pieces with the same size and shape.

Using a compass and straightedge to construct a reflection image of a figure. http://www.learner.org/courses/learningmath/geometry/session7/part_a/bisector.html

3.3 & 3.5 Rotations and Symmetry Common Core Standards CC.9-12.G.CO.2 CC.9-12.G.CO.5

3.3 Rotations A rotation r is a transformation that rotates each point in the preimage about a point P, called the center of rotation, by an angle measure x˚, called the angle of rotation. A rotation has these properties: 1) The image of P is Pˊ (that is, Pˊ = P). and (x˚, P)

2)For a preimage point A, PA = PAˊ and m<APAˊ = x˚. P = Pˊ

A rotation is a rigid motion, so length and angle measure are preserved. Note that a rotation is counterclockwise for a positive angle measure. A rotation is an isometry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180° or less.

S R R' T´ S' T' Rotate ∆RST clockwise 90° about the origin and name the coordinates of the new vertices. R(-2, 3) R‘(3, 2) S(0, 4) S‘(4, 0) T(1, 0) T‘(0, -1)

Neat way to rotate 90˚ about the origin Rotate 90 degrees clockwise (opp. of x then swap http://www.virtualnerd.com/tutorials/?id=PreAlg_09_01_0032 Rotate 270 counterclockwise degrees (opp. of y then swap) http://www.virtualnerd.com/tutorials/?id=PreAlg_09_01_0034

Rules for Rotations (clockwise & counterclockwise directions) 90 degrees (x, y) → (- y, x) (x, y) → (y, - x) 180 degrees (x, y) → (-x, - y) (x, y) → (-x, -y) 270 degrees (x, y) → (y, -x) (x, y) → (-y, x)

90˚ clockwise or 270˚ counterclockwise (x, y) (y, -x) 90˚ counterclockwise or 270˚ degrees clockwise (x, y) (-y, x) 180˚ clockwise or counterclockwise (x, y) (-x, -y)

a) 90° clockwise (b) 90° counterclockwise Name the coordinates of the vertices of the image after the given rotation of ∆ABC about the origin. a) 90° clockwise (b) 90° counterclockwise A(2, 0) A´(0, 2) B(2, 5) B´(-5, 2) C(6, 5) C´(-5, 6) Opp. Of y then swap. A (2, 0) A´(0, -2) B(2, 5) B´(5, -2) C(6, 5) C´(5, -6) Opp. Of x then swap. B C A

Ex: What is r ABCD. (90˚ rotation about the origin. O is the origin Ex: What is r ABCD? (90˚ rotation about the origin. O is the origin.) A(3, 5) Aˊ( ) B(1, 7) Bˊ( ) C(-2, 4) Cˊ( ) D(2, -1) Dˊ( ) Draw on the coordinate plane. Ex: What is r ABCD) (90˚, O) (- 270, O)

Steps in Rotating a Figure Use the following steps to draw the image of ΔABC after a 120° counterclockwise rotation about point P. 1. Draw a segment connecting vertex C and the center of rotation point P. A C P B

2. Use a protractor to measure a 120° angle counterclockwise and draw a ray. B P

3. Place the point of the compass at P and draw an arc from A to locate A´. B

4. Repeat Steps 1 – 3 for each vertex 4. Repeat Steps 1 – 3 for each vertex. Connect the vertices to form the image. P.

Theorem: Any rotation is a composition of reflections across two lines, which intersect at the center of rotation. The angle of rotation between two reflection lines is ½ the angle of rotation from the preimage and the image.

If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is a rotation about point P. The angle of rotation is 2x° (twice the angle x), where x° is the measure of the acute angle or right angle formed by k and m. k A A' m B B' 2x° x B'' P A''

Rotational Symmetry A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation of less than 180 degrees. A square has rotational symmetry because it maps onto itself by a rotation of 90 degrees. P 0° rotation 45° rotation 90° rotation r (45˚, P)

Rotations and Rotational Symmetry http://www.learner.org/courses/learningmath/geometry/session7/part_b/index.html

The type of symmetry for which there is rotation of 180˚ that maps a figure onto itself is called point symmetry. A parallelogram has 180˚ rotational symmetry, or point symmetry.

7.4 Translations and Vectors Common Core Standards CC.9-12.G.CO.2 CC.9-12.G.CO.4 CC.9-12.G.CO.5

3.2 Translations and Vectors

A translation is a transformation in a plane that maps all points of a preimage the same distance and in the same direction. The translation of ∆ABC by x units along the x-axis and by y units along the y-axis can be written as T (∆ABC) = ∆AˊBˊCˊ. A translation is a rigid motion, so length and angle measure are preserved. x, y

A translation has the following properties: AAˊ║ BBˊ║ CCˊ AAˊ BBˊ CCˊ ∆ABC and ∆AˊBˊCˊ have the same orientation.

You can find the image of a translation by gliding a figure in the plane.

Translations in a coordinate plane can be described by the following coordinate notation: (x, y) → (x + a, y + b) The notation uses an arrow to show how the transformation changes the coordinates of a general point, (x, y). Where a and b are constants. Each point shifts a units horizontally and b units vertically.

Example: Sketch a parallelogram with vertices R(-4, -1), S(-2, 0), T(-1, 3). Then sketch the image of the parallelogram after translation (x, y) → (x + 4, y – 2). or T (∆RST) = ∆RˊSˊTˊ 4, - 2

What are the vertices for the translation. T (STUV) What are the vertices for the translation? T (STUV)? (T stands for translation) S(-5, 6) T(3, 5) U(4, -2) V(-4, 1 ) - 1, - 4

A composition of a rigid motions is a transformation with two or more rigid motions in which the second rigid motion is performed on the image of the first rigid motion. (R ◦ T )(∆ABC) 1) Translate ∆ABC left 2 units and up 5 units. 2) Reflect ∆AˊBˊCˊ across line l . The small open circle indicates a composition of rigid motions on ∆ABC. l -2, 5

What is the composition of the transformation written as one transformation? T ◦ T (x, y) The translation T 3, - 2 1, - 1 4, -3

1. PP˝ is perpendicular to k and m. If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is a translation. If P˝ is the image of P, then the following is true: 1. PP˝ is perpendicular to k and m. 2. PP˝ = 2d, where d is the distance between k and m. m k Q˝ Q Q´ P´ P P˝ d

Reflect ∆ABC across line a. The image is ∆AˊBˊCˊ Reflect ∆ABC across line a. The image is ∆AˊBˊCˊ. Reflect ∆AˊBˊCˊ across line b is ∆A˝B˝C˝. If ∆ABC is translated 10 units to the right, its image is also ∆A˝B˝C˝. So, (R ◦ R )(∆ABC) = T (∆ABC). Notice that the distance between corresponding points on line a and line b is 5 units and BB˝ = AA˝ = CC˝ = 10 units. b a (10, 0)

Ex: Suppose n is the line with equation y = 1 Ex: Suppose n is the line with equation y = 1. Given ∆DEF with vertices D(0, 0), E(0, 3), and F(3, 0), what translation image is equivalent to (R ◦ R )(∆DEF)? n X-axis

Another way to describe a translation is by using a vector Another way to describe a translation is by using a vector. A vector is a quantity that has both direction (the arrowhead points in the direction) and magnitude (how long is it).

The diagram below shows a vector The diagram below shows a vector. The initial point, or starting point, of the vector is P and the terminal point, or ending point, is Q. The vector is named PQ, which is read as “vector PQ.” The horizontal component of PQ is 5 and the vertical component is 3. Component form: <5, 3> Q 3 units up P 5 units to the right

In each diagram, name each vector and write its component form. K N T S J M MN, <0, 4> TS, <3, -3> JK, <3, 4>

Example: The component form of GH is <4, 2>. Use GH to translate the triangle whose vertices are A(3, -1), B(1, 1), and C(3, 5).

3.4 Classification of Rigid Motions CC.9-12.G.CO.5

3.4 Glide Reflections and Compositions A translation, or glide, and a reflection can be performed one after the other to produce a transformation known as a glide reflection.

by the following steps: A translation maps P onto P’. A glide reflection is a transformation in which every point P is mapped onto a point P’’ by the following steps: A translation maps P onto P’. A reflection in a line k parallel to the direction of the translation maps P’ onto P’’. m k  A glide reflection is the composition of a reflection and a translation, where the line of reflection, m, is parallel to the directional vector line, v, of the translation. P’ P’’ v P

As long as the line of reflection is parallel to the direction of the translation, it does not matter whether you glide first and then reflect, or reflect 1st then glide.

Example: Use the information below to sketch the image of ΔABC after a glide reflection. A(-1, -3), B(-4, -1), C(-6, -4) Translation: (x, y) → (x + 10, y) Reflection: in the x-axis Same as: (R ◦ T )(∆ABC) = ∆AˊBˊCˊ X-axis <10, 0 >

In the previous example, reverse the order of the transformation In the previous example, reverse the order of the transformation. Notice that the resulting image will have the same coordinates as ∆AˊBˊCˊ. This is true because the line of reflection is parallel to the direction of the translation.

When two or more transformations are combined to produce a single transformation, the result is called a composition of the transformations. In a glide reflection, the order in which the transformation are performed does not affect the final image. For other compositions of transformations, the order may affect the final image.

Example: Sketch the image of PQ after a composition of the given rotation and reflection. P(2, -2), Q(3, -4) Rotation: 90 degrees counterclockwise about the origin Reflection: in the y-axis Same as: (R ◦ r )(PQ) Y-axis (90˚, O)

Repeat the previous problem, but switch the order of the composition by performing the reflection first and the rotation second. What do you notice? **A composition of isometries is always an isometry.

3.5 Symmetry

8.7 Dilations A dilation with center C and scale factor k is a transformation that maps every point P in the plane to a point P’ so that the following properties are true: 1. If P is not the center point C, then the image point P’ lies on CP. The scale factor k is a positive number such that k = CP’ / CP and k ≠ 1.

2. If P is the center point C, then P = P’. The dilation is a reduction if 0 < k < 1 and it is an enlargement if k > 1.

P 6 P’ 3 Q Q’ C R’ R Reduction: k = CP’ / CP = 3/6 = 1/2

P’ 5 P 2 R’ R C Q Q’ Enlargement: k = CP’ / CP = 5/2

Dilation in a Coordinate Plane Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4). Use the origin as the center and use a scale factor of ½. How does the perimeter of the preimage compare to the perimeter of the image?

Solution: Because the center of the dilation is the origin, you can find the image of each vertex by multiplying its coordinates by the scale factor. A(2, 2) →A’(1, 1) B(6, 2) →B’(3, 1) C(6, 4) →C’(3, 2) D(2, 4) →D’(1, 2) From the graph, you can see that the preimage has a perimeter of 12 and the image has a perimeter of 6. A preimage and its image after a dilation are similar figures. Therefore, the ratio of the perimeters of a preimage and its image is equal to the scale factor of the dilation.

Page 507 Drawing a Dilation

Identify the dilation and find its scale factor. a) (b) 3 7 A Q S T 1 B A’ R’ D Q’ S’ T’ B’ 3 D’ C Reduction or Enlargement? Scale factor? Reduction or Enlargement? Scale factor?

Answer to a: Enlargement, k = 7/3 Answer to b: Reduction, k = 3/4

Shadow puppets have been used in many countries for hundreds of years Shadow puppets have been used in many countries for hundreds of years. A flat figure is held between a light and a screen. The audience on the other side of the screen sees the puppet’s shadow. The shadow is a dilation, or enlargement, of the shadow puppet. When looking at a cross sectional view, ∆LCP ∆LSH. The shadow puppet shown is 12 inches tall (CP). Find the height f the shadow, SH, for each distance from the screen, In each case, by what percent is the shadow larger than the puppet? LC = LP = 59 in; LS = LH = 74 in: Set up prop. 59/74 = 12/x etc LC = LP = 66 in; LS = LH = 74 in S C Shadow Puppet shadow L P H

Answers: a) 59/74 = 12/SH SH = 15 inches Scale factor: SH/CP = 15/12 = 1.25 The shadow is 25% larger than the puppet. b) 66/74 = 12/SH SH = 13.45 inches Scale factor: SH/CP = 13.45/12 = 1.12 The shadow is 12% larger than the puppet. Note: As puppet moves closer to screen, the shadow height decreases.

Shadow puppets Pilobolus.wmv