2 Game Plan Distortions Orientations Parallel Path Translation Rotation Reflection
3 Game Plan Con’t Combination – Glide Reflection Combinations Single IsometrySimiltudes DilutationSeries of Tranformation
4 Transformation Transformation Formula Format (x, y) (a, b) The Any time a figure is moved in the plane we call this a transformation.As mathematicians we like to categorize these transformations.The first category we look at are the ugly ones or distortions Transformation FormulaFormat (x, y) (a, b) Theold x becomes aold y becomes b
5 Examples Eg (x,y) (x + y, x – y) A (2, -5) K (-4, 6) (-3, 7) A’ B (-1, 8) (-3, 7) A’(2, -10) K’(-59, 3) B’
6 Using a Graph Let’s try one on graph paper Consider A (1,4) B (7,2) C (3, –1)(x,y) (x + y, x – y)Step 1: Calculate the new pointsStep 2: Plot the points i.e A A’ B B’ etc.A (1,4) (5, -3) A’B (7,2) (9,5) B’C (3 – 1) (2,4) C’
7 Ex#1: Put on Graph Paper Formula Box (x,y) (x+y, x-y) C’Formula Box(x,y) (x+y, x-y)A (1,4) (5,-3) A’B (7,2) (9,5) B’C (3,–1) (2,4) C’BNotice, this graph is off the page… make sure yours does not CA’
8 Orientation To examine figures, we need to know how they line up. We are concerned withClockwise (CW)Counterclockwise (CCW)
9 Orientation Consistency is Key Start with A go ccw Eg A’ A B C B’ C’ Orientation ABC and A’ B’ C’Orientation is the same
10 Orientation Con’t A A’ B C B’ C’ What happened to the orientation? Orientation has changed
12 Parallel PathsWhen 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
13 Parallel PathWe say a transformation where all the vertices’ paths are parallel, the object has experienced a parallel pathA’AB’C’BCWe say line AA’ is a pathThese are a parallel path
14 It is called Intersecting Paths Parallel PathA’AC’B’BCThese are not parallel pathsIt is called Intersecting Paths
15 Parallel Path Solution: A + C BWhich two letters form a parallel path? If you choose A, must go with A’; B with B’ etc.B’A’CSolution: A + CDo stencil #1-3
16 IsometryIt is a transformation where a starting figure and the final figure are congruent.Congruent: equal in every aspect (side and angle)
17 Isometry Example Are these figures congruent? Since 16 = 24 = 32 KA1612624BC32T9PSince 16 = 24 = 328/3 = 8/3 = 8/3Are these figures congruent?
18 Translation Sometimes called a slide or glide Formula t (a,b) Means (x,y) (x + a, y + b)Eg t (-3,4) Eg Given A (7,1) B (3,5) C(4,-1)Draw t (-3,4) Include formula box and type box on graphType box means label and answer orientation (same / changed)Parallel Path (yes / no)
19 Type Box Orientation – same Parallel Path - yes Given A (7,1) B (3,5) C(4,-1) Draw t (-3,4)A’B’BFormula Box(x,y) (x-3, y+4)A (7,1) (4,5) A’B (3,5) (0,9) B’C (4,–1) (1,3) C’C’AType BoxCOrientation – sameParallel Path - yes
20 Rotation In theory we need a rotation point An angle A direction In practice – we use the origin as the rotation pointAngles of 90° and 180°Direction cw and ccwNote in math counterclockwise is positive
21 Rotation Formula r (0, v ) Rotation Origin Angle & Direction r (0, -90°) means a rotation about the origin 90° clockwise(x,y) (y, -x)When x becomes -x it changes sign. Thus – becomes +; + becomes –Notice the new position of x and y.
22 Rotationr (0, 90) means rotation about the origin 90° counterclockwise(x,y) (-y, x)r (0, 180) means rotation about the origin (direction does not matter)(x,y) (-x, -y)
23 Rotation Practice Given A (-4,2) B (-2,4) C (-5,5) Draw r (0,90); include formula box on graphYou try it on a graph!
24 r (0, 90) A (-4,2) B (-2,5) C(-5,-5) Orientation – same (x,y) (-y, x)A (-4,2) (-2,-4) A’B (-2,5) (-5,-2) B’C (-5,-5) (5,-5) C’AOrientation – sameParallel Path - noB’CC’A’
25 Reflections In theory we need a reflection line Sx = reflection over x axis(x,y) (x, -y)Sy = reflection over y axis(x,y) (-x,y)S reflection over y = x(x,y) (y,x)S reflection over y = -x(x,y) (-y,-x)
26 Memory Aid It is very important to put all these formulas on one page. P 160 #7 Put on separate sheetP161 #9You should be able to do all these transformation and understand how they work.
27 Combination NotationWhen we perform two or more transformations we use the symbol °It means afterA ° BMeans A after Bt (-3,2) ° Sy meansA translation after a reflection (you must start backwards!)
28 Combination Glide Reflection Draw t (-3,2) ° Sy(x,y) (-x,y) (x-3, y+2)A (4,3) C (-1,2) (1,2) C’ (-2,4) C’’Orientation changed, Parallel Path noWhat kind of isometry is this? It is a GLIDE REFLECTIONLet us look at the four types of isometries(-4,3) A’ (-7,5) A’’B (1,-3) (-1,-3) B’ (-4,-1) B’’
29 Single IsometryOrientation Same? Parallel Path?Any transformation in the plane that preserves the congruency can be defined by a single isometry.TRANSLATIONYESYESNoROTATIONYESREFLECTIONNoGLIDE REFLECTIONNo
30 Table Representation Orientation Same (maintained) Orientation Different (changed)With Parallel PathTranslationReflectionWithout Parallel PathRotationGlide Reflection
31 Similtudes & Dilitations 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 anglesThe similar shape we will create will have the same angle measurement and the sides will be proportional.The 1st part we need is this proportionality constant or scale factor.
32 Similtudes & Dilitations The 2nd 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 variationConsider transform ABC by a factor of 2 about point 0 (1,5).The scale factor is sometimes called k
33 Similtudes & Dilitations Sign of the scale factorPositive – both figures (original & new) are on the same side of pointNegative – both figures (original and new) are on the opposite sides of pointThe point is sometimes called the hole point