motion geometry part i

Motion GeometryMotion GeometryPart IPart I

Geometry

SolveProblems

Organize

Model

Compute

Communicate

Measure

Reason

Analyze

TransformatioTransformationsns

Transformations Transformations

A A transformation transformation is a change is a change in position, shape, or size of a in position, shape, or size of a figure.figure.

Example Example Putting together a jigsaw puzzles is Putting together a jigsaw puzzles is

an example motion geometry in an example motion geometry in action and can be used to illustrate action and can be used to illustrate transformations.transformations.

How does it work?How does it work? When you get a When you get a

new jigsaw puzzle, new jigsaw puzzle, you dump all the you dump all the pieces out of the pieces out of the box onto a table. box onto a table.

What do you do next?What do you do next? You probably turn the pieces over so You probably turn the pieces over so

that they are all face up. that they are all face up. You might adjust the angle of the pieces. You might adjust the angle of the pieces.

You might slide a piece across the table.You might slide a piece across the table.

Each of these represents a Each of these represents a transformation of the piece.transformation of the piece.

Each of these translations has Each of these translations has a special name.a special name.

Flipping the piece over is an example Flipping the piece over is an example of a of a reflection reflection (or flip).(or flip).

Changing the angle of the piece is an Changing the angle of the piece is an example of a example of a rotation rotation (or turn).(or turn).

Moving the piece across the table is Moving the piece across the table is an example of a an example of a translation translation (or (or slide).slide).

IsometryIsometry

If a figure and the figure formed by If a figure and the figure formed by transforming it are congruent, the transforming it are congruent, the transformation is called an transformation is called an isometryisometry. If a . If a transformation is an isometry, the transformation is an isometry, the sizesize and and shapeshape of the figure remains the same and of the figure remains the same and only the position of the figure changes. only the position of the figure changes.

FactFact In an isometry distance is also In an isometry distance is also

preserved. Since the figures before preserved. Since the figures before and after the transformation are and after the transformation are congruent, the distance between congruent, the distance between corresponding points does not corresponding points does not change.change.

What do you think? What do you think?

Is flipping a puzzle piece an Is flipping a puzzle piece an isometry?isometry?

SolutionSolution Yes. Yes. The image and object are congruent.The image and object are congruent. Shape, size, and distance are Shape, size, and distance are

preserved.preserved.

What do you think?What do you think? Is turning or rotating a puzzle piece Is turning or rotating a puzzle piece

an isometry?an isometry?

SolutionSolution Yes. Yes. The image and object are congruent.The image and object are congruent. Shape, size, and distance are Shape, size, and distance are

preserved.preserved.

What do you think?What do you think? Is sliding a puzzle piece across the Is sliding a puzzle piece across the

table an isometry?table an isometry?

SolutionSolution Yes. Yes. The image and object are congruent.The image and object are congruent. Shape, size, and distance are Shape, size, and distance are

preserved.preserved.

OrientationOrientationTheThe orientation orientation of an object refers of an object refers

to the order of its parts as you move to the order of its parts as you move around the object in a clockwise or a around the object in a clockwise or a counter-clockwise direction.counter-clockwise direction.

Example:Example:What is the orientation of the giraffe’s nose, What is the orientation of the giraffe’s nose, ears, and tail starting with the nose and going ears, and tail starting with the nose and going clockwise?clockwise?

SolutionSolution Nose – Ears – TailNose – Ears – Tail

What do you think?What do you think? If the giraffe is slid to a new position, If the giraffe is slid to a new position,

does its orientation change?does its orientation change?

Solution Solution No. It is still nose – ears – tail.No. It is still nose – ears – tail.

What do you think?What do you think? If the giraffe is turn or rotated, does If the giraffe is turn or rotated, does

its orientation change?its orientation change?

No. The orientation of the giraffe does not No. The orientation of the giraffe does not change. change. In both cases the order is nose – ears – tail.In both cases the order is nose – ears – tail.

What do you think?What do you think? If the giraffe is reflected, does its If the giraffe is reflected, does its

orientation change?orientation change?

Yes the orientation changes in a Yes the orientation changes in a reflection. Starting at the nose and going reflection. Starting at the nose and going clockwise, its orientation is now: nose – clockwise, its orientation is now: nose – tail – ears.tail – ears.

TranslationsTranslations

TranslationsTranslations

A A translationtranslation is a transformation is a transformation that moves all points of a figure the that moves all points of a figure the samesame distancedistance in the in the same directionsame direction..

Translations Translations In order to translate a figure you In order to translate a figure you need to know two things.need to know two things.

• How far will it be translated?How far will it be translated?• In what direction will it be In what direction will it be

translated? translated?

FactFactA translation (or slide) preservesA translation (or slide) preserves

size, size, shape, shape, distance, and distance, and orientation.orientation.

TerminologyTerminology

In a transformation, the given figure is In a transformation, the given figure is called the called the preimagepreimage and the transformed and the transformed figure is called the figure is called the imageimage. Points on the . Points on the image that correspond to points on the image that correspond to points on the preimage are labeled similarly but with preimage are labeled similarly but with primes. A transformation is said to primes. A transformation is said to mapmap a a figure onto its image.figure onto its image.

Try ItTry It1.1. Choose one of your attribute pieces.Choose one of your attribute pieces.2.2. Draw an arrow on your paper.Draw an arrow on your paper.3.3. Place your attribute piece at the end Place your attribute piece at the end

of the arrow. Trace around it.of the arrow. Trace around it.4.4. Use the arrow (vector) to represent Use the arrow (vector) to represent

the direction and distance, translate the direction and distance, translate your attribute piece. Trace around it. your attribute piece. Trace around it.

Did you align a vertex or a side at Did you align a vertex or a side at the foot of the arrow?the foot of the arrow?

Foot of arrowFoot of arrowFoot of arrowFoot of arrow

It is more difficult to translate using a It is more difficult to translate using a vertex than a side. You can slide the vertex than a side. You can slide the side along the arrow. BUTside along the arrow. BUT

B

A C

A'

B'

preimage

image

C'

u

Do not rotate as you slide.Do not rotate as you slide.If you rotate with the vertex alone it If you rotate with the vertex alone it

isisdifficult not to rotate as well as slide difficult not to rotate as well as slide

the the figure.figure.

uB

CA

Translating Polygons by Translating Polygons by ConstructionConstruction

A polygon can be translated by A polygon can be translated by translating its vertices and then translating its vertices and then connecting these points. connecting these points.

So it is only necessary to know how So it is only necessary to know how to translate a point in order to know to translate a point in order to know how to translate a polygon. how to translate a polygon.

ExampleExample Translate point A according to the given Translate point A according to the given

vector.vector.

PlanPlan1.1. Construct a parallelogram with the Construct a parallelogram with the

ends of the arrow (vector) and ends of the arrow (vector) and point A as three of its vertices.point A as three of its vertices.

2.2. The fourth vertex will be, the The fourth vertex will be, the required image. required image.

Use your compass to measure the Use your compass to measure the length of the vector length of the vector

Copy this length from point A in the Copy this length from point A in the general direction of the arrow.general direction of the arrow.

Using your compass measure the Using your compass measure the distance from the end of the arrow distance from the end of the arrow to point A. to point A.

Copy this distance from the head of Copy this distance from the head of the vector. the vector.

The intersection of arcs is the fourth The intersection of arcs is the fourth vertex.vertex.

Try ItTry It Draw a line segment on your paper and a Draw a line segment on your paper and a

vector (arrow) near it.vector (arrow) near it. Transform the segment according to the Transform the segment according to the

vector.vector.

Solution: Solution:

Application of a translationApplication of a translation

Frieze PatternsFrieze Patterns A frieze pattern is a pattern that A frieze pattern is a pattern that

repeats itself along a straight line. repeats itself along a straight line. The pattern may be mapped onto The pattern may be mapped onto itself with a translation. itself with a translation.

Wallpaper borders are practical Wallpaper borders are practical applications of frieze patterns. applications of frieze patterns.

Frieze patterns can be found around Frieze patterns can be found around the eaves of some old buildings. the eaves of some old buildings.

Translation with dot paperTranslation with dot paper Translations on dot paper can be Translations on dot paper can be

accomplished using the slope of the accomplished using the slope of the translation vector. translation vector.

Try It Try It

SolutionSolution

3

4preimage

image

Mathematical Notation of a Mathematical Notation of a TranslationTranslation

A translation, T, that moves an object h units A translation, T, that moves an object h units to the right or left and k units up or down is to the right or left and k units up or down is TT(h,k)(h,k)..

This may also be written using the following This may also be written using the following notation. T: (x, y) notation. T: (x, y) (x + h, y + k) (x + h, y + k)

If h is positive, the object moves to the right. If h is positive, the object moves to the right. If h is negative, the object moves to the left. If h is negative, the object moves to the left. If k is positive, the object moves up and If k is positive, the object moves up and If k is negative, the object moves down.If k is negative, the object moves down.

Try ItTry ItWhere would the point (2, Where would the point (2, 3) be 3) be

locatedlocatedafter the translation described by after the translation described by

TT((5,7)5,7)??

SolutionSolutionThe point moves left 5 and up 7 so The point moves left 5 and up 7 so TT((5, 7)5, 7) (2, (2,3) (2 3) (2 5, 5,3 + 7) 3 + 7)

The point moves from (2, The point moves from (2, 3) to (3) to (3, 4) 3, 4) under this translation.under this translation.

Try ItTry It Translate the triangle using TTranslate the triangle using T(3, (3, 4).4).

SolutionSolution

image

Try ItTry ItFind the preimage if the following Find the preimage if the following

image image resulted after the translation Tresulted after the translation T(5, (5, 3)3)..

image

SolutionSolution The flag was moved 5 units right and The flag was moved 5 units right and

3 units down. 3 units down. To undo this and return the flag to its To undo this and return the flag to its

original position, each point in the original position, each point in the flag must be moved 5 units left and 3 flag must be moved 5 units left and 3 units units up.up.

image

preimage

ReflectionsReflections

ReflectionsReflections If you look in a mirror you see your If you look in a mirror you see your

reflection. Your image looks like you reflection. Your image looks like you because it is the same size and because it is the same size and shape as you. The distance from shape as you. The distance from your nose to your lips is the same in your nose to your lips is the same in your reflection. However, when you your reflection. However, when you raise your right hand your image raise your right hand your image raise its left hand. raise its left hand.

Properties of ReflectionsProperties of ReflectionsA reflection preserves A reflection preserves

size,size, shape, andshape, and distance.distance.

It reverses orientation.It reverses orientation.

Image reflectorImage reflector An image reflector can be used to An image reflector can be used to

find the position and orientation of find the position and orientation of an object after it has been reflected. an object after it has been reflected.

Try ItTry ItUse your image reflector to Use your image reflector to complete the butterfly. complete the butterfly.

Place the beveled edge along the Place the beveled edge along the line of reflection. line of reflection.

Look through the reflector until you Look through the reflector until you see its image. see its image.

Trace the image.Trace the image.

The ButterflyThe ButterflyReflect the butterfly.Reflect the butterfly.

SolutionSolution

Exploration Exploration Find a point on the left half of your butterfly and Find a point on the left half of your butterfly and

mark that point and its image.mark that point and its image. Draw a line segment connecting the two points.Draw a line segment connecting the two points. Use your compass to compare the distance of Use your compass to compare the distance of

each point to the reflecting line. each point to the reflecting line. Use the corner of a piece of paper and test to Use the corner of a piece of paper and test to

see whether or not the line between the points see whether or not the line between the points is perpendicular to the line of reflection.is perpendicular to the line of reflection.

Pick another point on the butterfly and try the Pick another point on the butterfly and try the four steps above again. four steps above again.

Write a conjecture about a point and its Write a conjecture about a point and its reflection over a reflecting line. reflection over a reflecting line.

A Reflection A Reflection A A reflectionreflection is a transformation in which each is a transformation in which each

point is mapped onto to its image over a line in point is mapped onto to its image over a line in such a way that the line is the perpendicular such a way that the line is the perpendicular bisector of the line segment connecting the point bisector of the line segment connecting the point and its image.and its image.

ReflectionReflectionIn order to reflect a figure you need In order to reflect a figure you need only knowonly know

The position of the mirror or line of The position of the mirror or line of reflection.reflection.

FactFact A polygon can be reflected by A polygon can be reflected by

construction by reflecting its vertices construction by reflecting its vertices and then connecting the points. and then connecting the points.

So it is only necessary to know how So it is only necessary to know how to reflect a point in order to know to reflect a point in order to know how to reflect a polygon.how to reflect a polygon.

Reflect the point by Reflect the point by construction.construction.

SolutionSolution Drop a perpendicular from the point Drop a perpendicular from the point

to the line.to the line. Extend the perpendicular beyond Extend the perpendicular beyond

the line.the line. Measure the length of the point to Measure the length of the point to

the line and copy that length on the the line and copy that length on the other side of the reflecting line other side of the reflecting line along the perpendicular.along the perpendicular.

SolutionSolution

Try It Try It Reflect the triangle over the line of Reflect the triangle over the line of

reflection by construction. reflection by construction.

SolutionSolution

Try It Try It Find the line of reflection in the following Find the line of reflection in the following

figure.figure.

Solution Solution

Application Application A kaleidoscope is a device containing A kaleidoscope is a device containing

stationary mirrors and loose pieces of stationary mirrors and loose pieces of colored glass. colored glass.

The glass pieces are reflected many The glass pieces are reflected many times in the mirrors depending upon the times in the mirrors depending upon the number of mirrors and the angles at number of mirrors and the angles at which they are placed. which they are placed.

As the kaleidoscope is rotated, the pieces As the kaleidoscope is rotated, the pieces of glass move and an ever changing of glass move and an ever changing colorful, symmetric pattern is created. colorful, symmetric pattern is created.

How does it work?How does it work?

How does it work?How does it work? Not only are the objects placed Not only are the objects placed

between the mirrors reflected, but so between the mirrors reflected, but so are the objects in the virtual mirrors are the objects in the virtual mirrors that are created. that are created.

The angle between the mirrors is The angle between the mirrors is critical so that eventually reflections critical so that eventually reflections will coincide. The viewing eye piece will coincide. The viewing eye piece is usually circular.is usually circular.

How does it work?How does it work? If an object is placed between the If an object is placed between the

mirrors, it is reflected by both mirrors, it is reflected by both mirrors.mirrors. Original

Shape MirrorMirrorVirtual Mirror

Virtual Mirror

How does it work?How does it work? Second reflectionSecond reflection

How does it work?How does it work? Third reflectionThird reflection

How does it work?How does it work? Fourth reflectionFourth reflection

Change the angleChange the angle Suppose the angle is changed to 36Suppose the angle is changed to 3600.. How will it look after the reflections are How will it look after the reflections are

complete?complete?

SolutionSolution

Reflecting with dot paper.Reflecting with dot paper.To reflect objects on dot paper when the line To reflect objects on dot paper when the line of reflection has a slope of 1 or of reflection has a slope of 1 or 1:1:

Find the perpendicular from the preimage to Find the perpendicular from the preimage to the line of reflection by counting dots along the line of reflection by counting dots along the diagonal from the point to the line of the diagonal from the point to the line of reflection.reflection.

The image will be the same distance from The image will be the same distance from the line of reflection on the same diagonal the line of reflection on the same diagonal but on the other side of the line of reflection.but on the other side of the line of reflection.

Dot Paper Reflection Dot Paper Reflection Reflect point A over the given line of reflection by Reflect point A over the given line of reflection by

counting dots.counting dots.

A

Count the DotsCount the Dots The line of reflection has a slope of 1. Point A is 5 The line of reflection has a slope of 1. Point A is 5

diagonal units from the line of reflection so is 5 diagonal units from the line of reflection so is 5 diagonal units on the other side of the line of diagonal units on the other side of the line of reflection.reflection.

A1

5

4

3

21

3

4

5

2

A'

Try ItTry It Reflect the line segment.Reflect the line segment.

SolutionSolution Count the dots along the diagonal.Count the dots along the diagonal.

A B

A'

B'

Reflection over the y-axisReflection over the y-axis Reflect the points over the y-axisReflect the points over the y-axis

A

B

C D

y - axis

x - axis

Reflections over the y-axis Reflections over the y-axis Reflect the points over the y-axis.Reflect the points over the y-axis.

E

F

H

G

y- axis

x- axis

Find Coordinates Find Coordinates Complete the Complete the

chart for both chart for both the object the object and its and its image.image.

Point Coordinates of Preimage Coordinates of Image

A

B

C

D

E

F

G

H

SolutionSolutionPointPoint Coordinates of Coordinates of

PreimagePreimageCoordinates of Coordinates of

ImageImageAA ((8, 9)8, 9) (8, 9)(8, 9)BB (1, 6)(1, 6) ((1, 6)1, 6)CC (7,(7,4)4) ((7, 7, 4)4)DD ((5, 5, 6)6) (5, (5, 6)6)EE ((3, 5)3, 5) (3, 5)(3, 5)FF (4, 2)(4, 2) ((4, 2)4, 2)GG (3, (3, 7)7) ((3, 3, 7)7)HH ((5, 5, 2)2) (5, (5, 2)2)

Try ItTry It Write a conjecture giving the Write a conjecture giving the

coordinate of the image of point (x, coordinate of the image of point (x, y) reflected over the y - axis.y) reflected over the y - axis.

SolutionSolution The x-coordinate of the image is the The x-coordinate of the image is the

negative of the x-coordinate of the negative of the x-coordinate of the preimage. preimage.

The y-coordinate remains the same. The y-coordinate remains the same.

The coordinates of the image areThe coordinates of the image are((x, y).x, y).

Notation of ReflectionsNotation of Reflections The mathematical notation for a The mathematical notation for a

reflection is a lower case r with the reflection is a lower case r with the equation of the line of reflection or a equation of the line of reflection or a letter indicating an axis as a letter indicating an axis as a subscript. subscript.

To indicate a reflection over the y-To indicate a reflection over the y-axis either raxis either rx=0 x=0 or ror ryy is used. is used.

Notations of Reflections over Notations of Reflections over the the

y-axisy-axis rrx=0x=0(x, y) = ((x, y) = (x, y)x, y) rryy(x, y) = ((x, y) = (x, y)x, y)

Reflections of the x-axisReflections of the x-axis Reflect the same points over the x-Reflect the same points over the x-

axis.axis. Make a conjecture as to the Make a conjecture as to the

coordinates of (x, y) reflected over coordinates of (x, y) reflected over the x-axis.the x-axis.

SolutionSolution The x-coordinate of the image and The x-coordinate of the image and

the preimage are the same. the preimage are the same. The yThe y--coordinate of the image is the coordinate of the image is the

negative of the y-coordinate of the negative of the y-coordinate of the preimage. preimage.

The coordinates of the (x, The coordinates of the (x, y)y)

Notations of Reflections over Notations of Reflections over thethe

x-axisx-axis rry=0y=0(x, y) = (x, (x, y) = (x, y)y) rrxx(x, y) = (x, (x, y) = (x, y)y)

Reflections over the line Reflections over the line y=xy=x

Reflect the points over the line y=x.Reflect the points over the line y=x. Make a conjecture using your results.Make a conjecture using your results.

SolutionSolution The x and y coordinates are The x and y coordinates are

interchanged. interchanged. Therefore the reflected image of (x, Therefore the reflected image of (x,

y) is (y, x).y) is (y, x).

Notation of Reflections of the Notation of Reflections of the lineliney=xy=x

rry=xy=x(x,y) = (y, x)(x,y) = (y, x)

Use Patty Paper to ReflectUse Patty Paper to Reflect Draw figure to be reflected.Draw figure to be reflected. Draw line of reflection.Draw line of reflection. Fold paper at the line of reflection.Fold paper at the line of reflection. Copy figure on the other side of the Copy figure on the other side of the

fold.fold. Unfold.Unfold.

Try ItTry It Put a pencil in both hands. Put a pencil in both hands. If you are right-handed start with If you are right-handed start with

your hand together.your hand together. If you are left-handed start with your If you are left-handed start with your

hands apart.hands apart. Write you name with both hands at Write you name with both hands at

the same time.the same time.

Right-handedRight-handed

Left-handedLeft-handed

RotationsRotations

RotationsRotations A rotation is a rigid motion just as A rotation is a rigid motion just as

translations and reflections are. translations and reflections are. The figure that is rotated cannot The figure that is rotated cannot

bend or change shape. bend or change shape. The figure and its image are The figure and its image are

congruent under a rigid motion. congruent under a rigid motion. A rotation is an isometry. A rotation is an isometry.

RotationsRotations

A A rotationrotation is a transformation is a transformation thatthat moves each point on an object moves each point on an object a a given angle around a given given angle around a given point.point.

RotationsRotations Some fixed point in the plane is used Some fixed point in the plane is used

as the center of the rotation. as the center of the rotation. Every point in the figure is turned a Every point in the figure is turned a

given number of degrees about the given number of degrees about the point. point.

RotationsRotationsIn order to rotate an object you must In order to rotate an object you must

knowknow The center of the rotation.The center of the rotation. The angle of the rotation.The angle of the rotation. The direction of the rotation.The direction of the rotation.

Rotation ExplorationRotation Exploration In order to visualize a rotation, try In order to visualize a rotation, try

this. Trace the following letter F and this. Trace the following letter F and its “string” on a piece of tracing its “string” on a piece of tracing paper. paper.

Rotation ExplorationRotation Exploration Trace a circle with the radius of the Trace a circle with the radius of the

string. string. Place the letter F on the circle with the Place the letter F on the circle with the

other end of the string at point P, the other end of the string at point P, the center of the circle.center of the circle.

Move the F around the circle, keeping Move the F around the circle, keeping the end of the string on point P. the end of the string on point P.

Trace the F in several positions around Trace the F in several positions around the circle.the circle.

Rotation Exploration Rotation Exploration Your results will look something like Your results will look something like

this.this.

Notation for a RotationNotation for a Rotation The notation for a rotation is an upper The notation for a rotation is an upper

case script case script RR with two subscripts. with two subscripts. The first subscript names the point of The first subscript names the point of

rotation rotation The second subscript indicates the The second subscript indicates the

degree of the rotation. degree of the rotation. RRp, 90p, 90oo represents a rotation of 90 degrees about

point P.

Try It Try It Perform the rotation Perform the rotation RRAA, , ( (P)P) by by

construction. Label the image.construction. Label the image.

A

P

SolutionSolution Point A is the center of the rotation. Point A is the center of the rotation. Point P is the point to be rotated. Point P is the point to be rotated. Connect A and P with a line segment.Connect A and P with a line segment. Copy angle Copy angle with vertex at A and with vertex at A and

segment AP as the initial side of the segment AP as the initial side of the angle. angle.

Copy the angle in the counterclockwise Copy the angle in the counterclockwise direction as indicated. direction as indicated.

SolutionSolution

P' P

A

Try It Try It Rotate Rotate RRPP,, AB by construction. AB by construction. Label the image.Label the image. B

A

β

SolutionSolution

β

A

B

A'

B'

P

β

β

Dot Paper Rotations Dot Paper Rotations Rotate a point A Rotate a point A 909000 (clockwise) (clockwise)

around point P. Call it point C. around point P. Call it point C. Rotate a point A 90Rotate a point A 9000 (counter- (counter-

clockwise) around point P. Call it clockwise) around point P. Call it point CC.point CC.

Try It Try It Use slopes to rotate the point A .Use slopes to rotate the point A .

P

A

Solution Solution The slope of the line segment between A The slope of the line segment between A

and P is 3/2. and P is 3/2. The slope of the segment between C and P The slope of the segment between C and P

and between CC and P must be negative and between CC and P must be negative 2/3. 2/3.

The only difference is the direction of the The only difference is the direction of the rotation. Use rotation. Use 2/3 or 2/(2/3 or 2/(3) to find the 3) to find the points.points.

SolutionSolution

P C

A CC

Try It Try It Rotate the line segment 90 degrees Rotate the line segment 90 degrees

about point P.about point P.

A

B

P

SolutionSolution

B' B

P A'

A

Coordinate GeometryCoordinate GeometryUse coordinate geometry to make the Use coordinate geometry to make the following rotations.following rotations. RR(0,0), 90(0,0), 9000 (2, 3)(2, 3) RR((1, 1, 2), 2), 909000 (2, 3)(2, 3)

SolutionSolution The first image is at (The first image is at (3, 2).3, 2).

SolutionSolution The second image is at (4, The second image is at (4, 5).5).

The EndThe End