Transformational Geometry Math 314

Game Plan Distortions Orientations Parallel Path Translation Rotation Reflection

Game Plan Con’t Combination – Glide Reflection Combinations Single Isometry Similtudes Dilutation Series of Tranformation

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 Formula Format (x, y)  (a, b) The old x becomes a old y becomes b

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’

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 points Step 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’

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’ B Notice, this graph is off the page… make sure yours does not  C A’

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

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

Orientation Con’t A A’ B C B’ C’ What happened to the orientation? Orientation has changed

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

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

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

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

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

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

Isometry Example Are these figures congruent? Since 16 = 24 = 32 K A 16 12 6 24 B C 32 T 9 P Since 16 = 24 = 32 6 9 12 8/3 = 8/3 = 8/3 Are these figures congruent?

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 graph Type box means label and answer orientation (same / changed) Parallel Path (yes / no)

Type Box Orientation – same Parallel Path - yes Given A (7,1) B (3,5) C(4,-1) Draw t (-3,4) A’ B’ B Formula 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’ A Type Box C Orientation – same Parallel Path - yes

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

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.

Rotation r (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)

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

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’ A Orientation – same Parallel Path - no B’ C C’ A’

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)

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

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

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 no What kind of isometry is this? It is a GLIDE REFLECTION Let us look at the four types of isometries (-4,3) A’  (-7,5) A’’ B (1,-3)  (-1,-3)  B’ (-4,-1) B’’

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

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

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 angles The 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.

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 variation Consider transform ABC by a factor of 2 about point 0 (1,5). The scale factor is sometimes called k

Similtudes & Dilitations Sign of the scale factor 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 The point is sometimes called the hole point

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

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