11-19 11-19 S 6.7S 6.7: : Perform Similarity Perform Similarity TransformationsTransformations

Review:Review:Transformations: Transformations: when a when a geometric figure is geometric figure is moved or changed in moved or changed in some way to produce a some way to produce a new figurenew figure

ImageImage is what the is what the newnew figure is called.figure is called.

Congruence TransformationsCongruence Transformationschange the position of a change the position of a figure without changing its figure without changing its size or shape.size or shape.

Three Types of the aboveThree Types of the above:: TranslationTranslation ReflectionReflection RotationRotation

Translation: Translation: every point on every point on the figure is moved the the figure is moved the same direction and same direction and distancedistance

(think of it as (think of it as ““slidingsliding””))

Coordinate Notation for a translation:(x,y) (x+a, y+b) Ex. (x,y) (x + 5, y + 7)Each point (x,y) of the blue figure is translated horizontally a units and vertically b units Ex. (-5, -6) (0, 1) (same for other 2 vertices)

a

b

x

y

Same orientation, just different position

Reflection:Reflection: a line of a line of reflection is used to reflection is used to create a create a mirror imagemirror image of of the original figurethe original figure

(think of it as (think of it as ““flippingflipping””))

Multiply the y coordinate by -1Multiply the y coordinate by -1(x,y) (x,-y)(x,y) (x,-y)

Ex: (2, 5) (2, -5)Ex: (2, 5) (2, -5)

Multiply the x coordinate by -1Multiply the x coordinate by -1(x,y) (-x,y)(x,y) (-x,y)

Ex: (1, 6) (-1, 6)Ex: (1, 6) (-1, 6)

Reflection in the x-Reflection in the x-axisaxis

Reflection in the y-Reflection in the y-axisaxisy

x

y

x

(x,y)

(x,-y)

(-x,y)(x,y)

Coordinate Notation for a Reflection

Rotation: Rotation: a figure is turned a figure is turned about a fixed point called about a fixed point called thethe center of rotation center of rotation

(think of it as (think of it as ““turning in a turning in a circlecircle””))

90 clockwise 90 clockwise RotationRotation

60 60 counterclockwise counterclockwise rotationrotationy y

x x

Rotation: need direction of rotation and degrees

Vocabulary:Vocabulary:Dilation: Dilation: a transformation a transformation that stretches (enlarges) or that stretches (enlarges) or shrinks (reduces) a figure shrinks (reduces) a figure to create a to create a similarsimilar figure figure

If enlarged, called an If enlarged, called an enlargement.enlargement.

If reduced, called a If reduced, called a reduction.reduction.

Center of Dilation:Center of Dilation:The fixed point from which The fixed point from which the figure is enlarged or the figure is enlarged or reducedreduced

(called (called ““with respect with respect to . . .to . . .””; usually the origin); usually the origin)

Scale Factor of a Scale Factor of a Dilation:Dilation:

Ratio of the side length Ratio of the side length of the image to the of the image to the corresponding side corresponding side length of the original length of the original figurefigure

Coordinate Notation for a Coordinate Notation for a DilationDilation

With respect to the origin,With respect to the origin,(x, y) (kx, ky) where k is (x, y) (kx, ky) where k is

the the scale scale factorfactor

Reduction: k is < 1 , but > 0.Reduction: k is < 1 , but > 0.Enlargement: k > 1Enlargement: k > 1

How to Draw a DilationHow to Draw a Dilation (Steps)- (Steps)-p.408p.408

1.1. Plot Plot the vertices given and the vertices given and connect with straight lines to connect with straight lines to create a geometric figure. create a geometric figure.

2. Draw rays from the origin 2. Draw rays from the origin through the vertices given that through the vertices given that extend well beyond the vertices.extend well beyond the vertices.

3. Open your compass the distance 3. Open your compass the distance from the origin to one vertex.from the origin to one vertex.

4. Keeping the compass open that 4. Keeping the compass open that same distance, mark an arc on same distance, mark an arc on the ray that starts at the vertex.the ray that starts at the vertex.

5. The intersection of the ray and 5. The intersection of the ray and the arc mark is a vertex on the the arc mark is a vertex on the image.image.

6. Do the same for the other 6. Do the same for the other vertices and their rays.vertices and their rays.

S 6.7, Ex. 1-2, GP 1-2.ppt

(If absent, insert examples 1 & 2 (If absent, insert examples 1 & 2 from your textbook, on graph from your textbook, on graph paper, in your notes, with solution paper, in your notes, with solution shown)shown)

Finding the scale factor and proving a Finding the scale factor and proving a figure is a dilation.figure is a dilation. The change from The change from each vertex of the original figure to each vertex of the original figure to its corresponding vertex of the image its corresponding vertex of the image must be by the same scale factor.must be by the same scale factor.

S 6.7 Ex 3-4, GP 3-4.ppt(If absent, put examples 3 & 4 in your (If absent, put examples 3 & 4 in your

notes from the textbook. Use graph notes from the textbook. Use graph paper slips if appropriate.)paper slips if appropriate.)