UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS …lilliepad.pbworks.com/w/file/fetch/70091129/AG Walch U1 L1Pt1... · ... dilated passes through the center of dilation, then the image

UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFSLesson 1: Investigating Properties of Dilations

Instruction

CCGPS Analytic Geometry Teacher Resource U1-10

© Walch Education

IntroductionThink about resizing a window on your computer screen. You can stretch it vertically, horizontally, or at the corner so that it stretches both horizontally and vertically at the same time. These are non-rigid motions. Non-rigid motions are transformations done to a figure that change the figure’s shape and/or size. These are in contrast to rigid motions, which are transformations to a figure that maintain the figure’s shape and size, or its segment lengths and angle measures.

Specifically, we are going to study non-rigid motions of dilations. Dilations are transformations in which a figure is either enlarged or reduced by a scale factor in relation to a center point.

Key Concepts

• Dilations require a center of dilation and a scale factor.

• The center of dilation is the point about which all points are stretched or compressed.

• The scale factor of a figure is a multiple of the lengths of the sides from one figure to the transformed figure.

• Side lengths are changed according to the scale factor, k.

• The scale factor can be found by finding the distances of the sides of the preimage in relation to the image.

• Use a ratio of corresponding sides to find the scale factor: length of image side

length of preimage sidescale factor=

• The scale factor, k, takes a point P and moves it along a line in relation to the center so that •k CP CP= ′ .

Prerequisite Skills

This lesson requires the use of the following skills:

• operating with fractions, including complex fractions

• operating with decimals

• calculating slope

• determining parallel lines


Instruction

CCGPS Analytic Geometry Teacher Resource © Walch EducationU1-11

C

P

P

P is under a dilation of scale factor k through center C.

k • CP = CP

• Ifthescalefactorisgreaterthan1,thefigureisstretchedormadelargerandiscalledanenlargement.(Atransformationinwhichafigurebecomeslargerisalsocalledastretch.)

• Ifthescalefactorisbetween0and1,thefigureiscompressedormadesmallerandiscalledareduction.(Atransformationinwhichafigurebecomessmallerisalsocalledacompression.)

• Ifthescalefactorisequalto1,thepreimageandimagearecongruent.Thisiscalledacongruency transformation.

• Anglemeasuresarepreservedindilations.

• Theorientationisalsopreserved.

• Thesidesofthepreimageareparalleltothecorrespondingsidesoftheimage.

• Thecorresponding sides are thesidesoftwofiguresthatlieinthesamepositionrelativetothefigures.

• Intransformations,thecorrespondingsidesarethepreimageandimagesides,so AB andA B′ ′ arecorrespondingsidesandsoon.

• ThenotationofadilationinthecoordinateplaneisgivenbyDk(x,y)=(kx,ky).Thescalefactor

ismultipliedbyeachcoordinateintheorderedpair.

• Thecenterofdilationisusuallytheorigin,(0,0).


Instruction


© Walch Education

• If a segment of the figure being dilated passes through the center of dilation, then the image segment will lie on the same line as the preimage segment. All other segments of the image will be parallel to the corresponding preimage segments.

• The corresponding points in the preimage and image are collinear points, meaning they lie on the same line, with the center of dilation.

C

T

T'

∆T'U'V' is ∆TUV under a dilation of scale factor k about center C.

U

V

V'

U'

Properties of Dilations

1. Shape, orientation, and angles are preserved.

2. All sides change by a single scale factor, k.

3. The corresponding preimage and image sides are parallel.

4. The corresponding points of the figure are collinear with the center of dilation.

Common Errors/Misconceptions

• forgetting to check the ratio of all sides from the image to the preimage in determining if a dilation has occurred

• inconsistently setting up the ratio of the side lengths

• confusing enlargements with reductions and vice versa


Instruction


Example 1

Is the following transformation a dilation? Justify your answer using the properties of dilations.

C

E (4, 2)

F(2, –2)

F (4, –4)

D(–2, 2)

E (2, 1)

D(–4, 4)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

y

x

1. Verify that shape, orientation, and angles have been preserved from the preimage to the image.

Both figures are triangles in the same orientation.

D D

E E

F F

∠ ≅∠ ′∠ ≅∠ ′∠ ≅∠ ′

The angle measures have been preserved.

Guided Practice 1.1.1


Instruction


© Walch Education

2. Verify that the corresponding sides are parallel.

�

�

(2 1)

( 2 2)

1

4

1

4= =

−− −

=−

=−my

xDE and �

�

(4 2)

( 4 4)

2

8

1

4= =

−− −

=−

=−′ ′my

xD E ;

therefore, DE D E′ ′ .

By inspection, EF E F′ ′ because both lines are vertical; therefore, they have the same slope and are parallel.

�

�

[2 ( 2)]

( 2 2)

4

41= =

− −− −

=−

=−my

xDF and �

�

[4 ( 4)]

( 4 4)

8

81= =

− −− −

=−

=−′ ′my

xD F ;

therefore, DF D F′ ′ . In fact, these two segments, DF and D F′ ′ , lie on the same line.

All corresponding sides are parallel.

3. Verify that the distances of the corresponding sides have changed by a common scale factor, k.

We could calculate the distances of each side, but that would take a lot of time. Instead, examine the coordinates and determine if the coordinates of the vertices have changed by a common scale factor.

The notation of a dilation in the coordinate plane is given by D

k(x, y) = (kx, ky).

Divide the coordinates of each vertex to determine if there is a common scale factor.

( 2,2) ( 4, 4)

4

22;

4

22

D D

x

x

y

yD

D

D

D

− → ′ −

=−−

= = =′ ′

(2,1) (4,2)

4

22;

2

12

E E

x

x

y

yE

E

E

E

→ ′

= = = =′ ′

(2, 2) (4, 4)

4

22;

4

22

F F

x

x

y

yF

F

F

F

− → ′ −

= = =−−

=′ ′

Each vertex’s preimage coordinate is multiplied by 2 to create the corresponding image vertex. Therefore, the common scale factor is k = 2.


Instruction


4. Verify that corresponding vertices are collinear with the center of dilation, C.

C

E (4, 2)

F(2, –2)

F (4, –4)

D(–2, 2)

E (2, 1)

D(–4, 4)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

y

x

A straight line can be drawn connecting the center with the corresponding vertices. This means that the corresponding vertices are collinear with the center of dilation.

5. Draw conclusions.

The transformation is a dilation because the shape, orientation, and angle measures have been preserved. Additionally, the size has changed by a scale factor of 2. All corresponding sides are parallel, and the corresponding vertices are collinear with the center of dilation.


Instruction


© Walch Education

Example 2

Is the following transformation a dilation? Justify your answer using the properties of dilations.

C

T(0, 5)

V(9, 0)

U(9, 5)

V (6, 0)

U(6, 5)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

y

x

1. Verify that shape, orientation, and angles have been preserved from the preimage to the image.

The preimage and image are both rectangles with the same orientation. The angle measures have been preserved since all angles are right angles.


Instruction


2. Verify that the corresponding sides are parallel.

TU ′ is on the same line as TU ; therefore, TU TU ′ .

CV ′ is on the same line as CV ; therefore, CV CV ′ .

By inspection, UV and U V′ ′ are vertical; therefore, UV U V′ ′ .

TC remains unchanged from the preimage to the image.

All corresponding sides are parallel.

3. Verify that the distances of the corresponding sides have changed by a common scale factor, k.

Since the segments of the figure are on a coordinate plane and are either horizontal or vertical, find the distance by counting.

In TUVC : In TU V C′ ′ :TU = VC = 9 6TU V C′ = ′ = UV = CT = 5 5U V CT′ ′ = =

The formula for calculating the scale factor is:

scale factor =length of image side

length of preimage side

Start with the horizontal sides of the rectangle.

6

9

2

3

TU

TU

′= = 6

9

2

3

V C

VC

′= =

Both corresponding horizontal sides have a scale factor of 2

3.

Next, calculate the scale factor of the vertical sides.

5

51

U V

UV

′ ′= = 5

51

CT

CT= =

Both corresponding vertical sides have a scale factor of 1.


Instruction


© Walch Education

4. Draw conclusions.

The vertical corresponding sides have a scale factor that is not

consistent with the scale factor of 2

3 for the horizontal sides. Since all

corresponding sides do not have the same common scale factor,

the transformation is NOT a dilation.

Example 3

The following transformation represents a dilation. What is the scale factor? Does this indicate enlargement, reduction, or congruence?

A'

B'

C'

A

B

C

12

15

10

3.75

2.5

3

1. Determine the scale factor.

Start with the ratio of one set of corresponding sides.

scale factor =length of image side

length of preimage side

2.5

10

1

4

A B

AB

′ ′= =

The scale factor appears to be 1

4.


Instruction


2. Verify that the other sides maintain the same scale factor.

3.75

15

1

4

B C

BC

′ ′= = and

3

12

1

4

C A

CA

′ ′= = . Therefore,

1

4

A B

AB

B C

BC

C A

CA

′ ′=

′ ′=

′ ′= and the scale factor, k, is

1

4.

3. Determine the type of dilation that has occurred.

If k > 1, then the dilation is an enlargement.

If 0 < k < 1, then the dilation is a reduction.

If k = 1, then the dilation is a congruency transformation.

Since 1

4k = , k is between 0 and 1, or 0 < k < 1.

The dilation is a reduction.

Documents

UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS …lilliepad.pbworks.com/w/file/fetch/70091129/AG Walch U1 L1Pt1... · ... dilated passes through the center of dilation, then the image