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Transformational Geometry Math 314

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Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation Rotation Reflection Reflection

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Game Plan Cont Combination – Glide Reflection Combination – Glide Reflection Combinations Combinations Single Isometry Single Isometry Similtudes Dilutation Similtudes Dilutation Series of Tranformation Series of Tranformation

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Transformation Any time a figure is moved in the plane we call this a transformation. Any time a figure is moved in the plane we call this a transformation. As mathematicians we like to categorize these transformations. As mathematicians we like to categorize these transformations. The first category we look at are the ugly ones or distortions The first category we look at are the ugly ones or distortions Transformation Formula Transformation Formula Format (x, y) (a, b) The Format (x, y) (a, b) The old x becomes a old y becomes b old y becomes b

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Examples Eg (x,y) (x + y, x – y) Eg (x,y) (x + y, x – y) A (2, -5) A (2, -5) K (-4, 6) K (-4, 6) Eg #2 (x,y) (3x – 7y, 2x + 5) Eg #2 (x,y) (3x – 7y, 2x + 5) B (-1, 8) B (-1, 8) (-3, 7) A (2, -10) K (-59, 3) B

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Using a Graph Lets try one on graph paper Lets try one on graph paper Consider A (1,4) B (7,2) C (3, –1) Consider A (1,4) B (7,2) C (3, –1) (x,y) (x + y, x – y) (x,y) (x + y, x – y) Step 1: Calculate the new points Step 1: Calculate the new points Step 2: Plot the points i.e A A B B etc. Step 2: Plot the points i.e A A B B etc. A (1,4) (5, -3) A A (1,4) (5, -3) A B (7,2) (9,5) B B (7,2) (9,5) B C (3 – 1) (2,4) C C (3 – 1) (2,4) C

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Ex#1: Put on Graph Paper AC B A B C Notice, this graph is off the page… make sure yours does not (x,y) (x+y, x-y) A (1,4) (5,-3) A B (7,2) (9,5) B C (3,–1) (2,4) C Formula Box

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Orientation To examine figures, we need to know how they line up. To examine figures, we need to know how they line up. We are concerned with We are concerned with Clockwise (CW) Counterclockwise (CCW)

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Orientation Consistency is Key Consistency is Key Start with A go ccw Start with A go ccw Eg Eg A CBCB Orientation ABC and A B C Orientation is the same A

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Orientation Cont A B C CB What happened to the orientation? Orientation has changed A

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Orientation Vocabulary Orientation the same… or Orientation the same… or preserved preserved unchanged unchanged constant constant Orientation changed or Orientation changed or not preserved not preserved changed changed not constant not constant

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Parallel Paths When we move or transform an object, we are interested in the path the object takes. To look at that we focus on paths taken by the vertices When we move or transform an object, we are interested in the path the object takes. To look at that we focus on paths taken by the vertices

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Parallel Path These are a parallel path A C C B B A We say line AA is a path We say a transformation where all the vertices paths are parallel, the object has experienced a parallel path

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Parallel Path A B C B A C These are not parallel paths It is called Intersecting Paths

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Parallel Path A B C A C B Which two letters form a parallel path? If you choose A, must go with A; B with B etc. Solution: A + C Do stencil #1-3

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Isometry It is a transformation where a starting figure and the final figure are congruent. It is a transformation where a starting figure and the final figure are congruent. Congruent: equal in every aspect (side and angle) Congruent: equal in every aspect (side and angle)

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Isometry Example CB A P T K Since 16 = 24 = /3 = 8/3 = 8/3 Are these figures congruent?

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Translation Sometimes called a slide or glide Sometimes called a slide or glide Formula t (a,b) Formula t (a,b) Means (x,y) (x + a, y + b) Means (x,y) (x + a, y + b) Eg t (-3,4) Eg t (-3,4) Eg Given A (7,1) B (3,5) C(4,-1) Eg Given A (7,1) B (3,5) C(4,-1) Draw t (-3,4) Include formula box and type box on graph Draw t (-3,4) Include formula box and type box on graph Type box means label and answer orientation (same / changed) Type box means label and answer orientation (same / changed) Parallel Path (yes / no) Parallel Path (yes / no)

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Given A (7,1) B (3,5) C(4,-1) Draw t (-3,4) Given A (7,1) B (3,5) C(4,-1) Draw t (-3,4) A C B A B C (x,y) (x-3, y+4) A (7,1) (4,5) A B (3,5) (0,9) B C (4,–1) (1,3) C Formula Box Type Box Orientation – same Parallel Path - yes

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Rotation In theory we need a rotation point In theory we need a rotation point An angle An angle A direction A direction In practice – we use the origin as the rotation point In practice – we use the origin as the rotation point Angles of 90° and 180° Angles of 90° and 180° Direction cw and ccw Direction cw and ccw Note in math counterclockwise is positive Note in math counterclockwise is positive

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Rotation Formula Formula r (0, v ) r (0, v ) Rotation Origin Angle & Direction Rotation Origin Angle & Direction r (0, -90°) means a rotation about the origin 90° clockwise r (0, -90°) means a rotation about the origin 90° clockwise (x,y) (y, -x) (x,y) (y, -x) When x becomes -x it changes sign. Thus – becomes +; + becomes – When x becomes -x it changes sign. Thus – becomes +; + becomes – Notice the new position of x and y. Notice the new position of x and y.

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Rotation r (0, 90) means rotation about the origin 90° counterclockwise r (0, 90) means rotation about the origin 90° counterclockwise (x,y) (-y, x) (x,y) (-y, x) r (0, 180) means rotation about the origin (direction does not matter) r (0, 180) means rotation about the origin (direction does not matter) (x,y) (-x, -y) (x,y) (-x, -y)

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Rotation Practice Given A (-4,2) B (-2,4) C (-5,5) Given A (-4,2) B (-2,4) C (-5,5) Draw r (0,90); include formula box on graph Draw r (0,90); include formula box on graph You try it on a graph! You try it on a graph!

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r (0, 90) A (-4,2) B (-2,5) C(-5,-5) A C B A B C (x,y) (-y, x) A (-4,2) (-2,-4) A B (-2,5) (-5,-2) B C (-5,-5) (5,-5) C Orientation – same Parallel Path - no

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Reflections In theory we need a reflection line In theory we need a reflection line S x = reflection over x axis S x = reflection over x axis (x,y) (x, -y) (x,y) (x, -y) S y = reflection over y axis S y = reflection over y axis (x,y) (-x,y) (x,y) (-x,y) S reflection over y = x S reflection over y = x (x,y) (y,x) (x,y) (y,x) S reflection over y = -x S reflection over y = -x (x,y) (-y,-x) (x,y) (-y,-x)

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Memory Aid It is very important to put all these formulas on one page. It is very important to put all these formulas on one page. P 160 #7 Put on separate sheet P 160 #7 Put on separate sheet P161 #9 P161 #9 You should be able to do all these transformation and understand how they work. You should be able to do all these transformation and understand how they work.

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Combination Notation When we perform two or more transformations we use the symbol ° When we perform two or more transformations we use the symbol ° It means after It means after A ° B A ° B Means A after B Means A after B t (-3,2 ) ° S y means t (-3,2 ) ° S y means A translation after a reflection (you must start backwards!) A translation after a reflection (you must start backwards!)

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Combination Glide Reflection Draw t (-3,2) ° S y Draw t (-3,2) ° S y (x,y) (-x,y) (x-3, y+2) (x,y) (-x,y) (x-3, y+2) A (4,3) A (4,3) C (-1,2) (1,2) C (-2,4) C C (-1,2) (1,2) C (-2,4) C Orientation changed, Parallel Path no Orientation changed, Parallel Path no What kind of isometry is this? It is a GLIDE REFLECTION What kind of isometry is this? It is a GLIDE REFLECTION Let us look at the four types of isometries Let us look at the four types of isometries (-4,3) A (-7,5) A B (1,-3) (-1,-3) B (-4,-1) B

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Single Isometry Any transformation in the plane that preserves the congruency can be defined by a single isometry. Any transformation in the plane that preserves the congruency can be defined by a single isometry. Orientation Same? Parallel Path? YES No TRANSLATION ROTATION REFLECTION GLIDE REFLECTION

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Table Representation Orientation Same (maintained) Orientation Different (changed) With Parallel Path TranslationReflection Without Parallel Path Rotation Glide Reflection

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Similtudes & Dilitations When a transformation changes the size of an object but not its shape, we say it is a similtude or a dilitation. When a transformation changes the size of an object but not its shape, we say it is a similtude or a dilitation. Note – we observe size by side length and shape by angles Note – we observe size by side length and shape by angles The similar shape we will create will have the same angle measurement and the sides will be proportional. The similar shape we will create will have the same angle measurement and the sides will be proportional. The 1 st part we need is this proportionality constant or scale factor. The 1 st part we need is this proportionality constant or scale factor.

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Similtudes & Dilitations The 2 nd part we need is a point from which this increase or decrease in size will occur. The 2 nd part we need is a point from which this increase or decrease in size will occur. Note – this is an exercise in measuring so there can be some variation Note – this is an exercise in measuring so there can be some variation Consider transform ABC by a factor of 2 about point 0 (1,5). Consider transform ABC by a factor of 2 about point 0 (1,5). The scale factor is sometimes called k The scale factor is sometimes called k

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Similtudes & Dilitations Sign of the scale factor Sign of the scale factor Positive – both figures (original & new) are on the same side of point Positive – both figures (original & new) are on the same side of point Negative – both figures (original and new) are on the opposite sides of point Negative – both figures (original and new) are on the opposite sides of point The point is sometimes called the hole point The point is sometimes called the hole point

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h ((1,5),2) A C B A B C mOA=2moA=2x2=4

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Other Examples P23 Example #8 P23 Example #8 P24 Spider Web P24 Spider Web Discuss scale factor Discuss scale factor Beam or light beam method

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