UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS Lesson...

UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFSLesson 4: Exploring Congruence

Instruction

CCGPS Analytic Geometry Teacher ResourceU1-242

© Walch Education

IntroductionRigid motions can also be called congruency transformations. A congruency transformation moves a geometric figure but keeps the same size and shape. Preimages and images that are congruent are also said to be isometries. If a figure has undergone a rigid motion or a set of rigid motions, the preimage and image are congruent. When two figures are congruent, they have the same shape and size. Remember that rigid motions are translations, reflections, and rotations. Non-rigid motions are dilations, stretches, and compressions. Non-rigid motions are transformations done to a figure that change the figure’s shape and/or size.

Key Concepts

• To decide if two figures are congruent, determine if the original figure has undergone a rigid motion or set of rigid motions.

• If the figure has undergone only rigid motions (translations, reflections, or rotations), then the figures are congruent.

• If the figure has undergone any non-rigid motions (dilations, stretches, or compressions), then the figures are not congruent. A dilation uses a center point and a scale factor to either enlarge or reduce the figure. A dilation in which the figure becomes smaller can also be called a compression.

• A scale factor is a multiple of the lengths of the sides from one figure to the dilated figure. The scale factor remains constant in a dilation.

• If the scale factor is larger than 1, then the figure is enlarged.

• If the scale factor is between 0 and 1, then the figure is reduced.

• To calculate the scale factor, divide the length of the sides of the image by the lengths of the sides of the preimage.

Prerequisite Skills

This lesson requires the use of the following skills:

• recognizing rotations, reflections, and translations

• setting up ratios

• using the Pythagorean Theorem


Instruction

CCGPS Analytic Geometry Teacher Resource© Walch EducationU1-243

• A vertical stretch or compression preserves the horizontal distance of a figure, but changes the vertical distance.

• A horizontal stretch or compression preserves the vertical distance of a figure, but changes the horizontal distance.

• To verify if a figure has undergone a non-rigid motion, compare the lengths of the sides of the figure. If the sides remain congruent, only rigid motions have been performed.

• If the side lengths of a figure have changed, non-rigid motions have occurred.

Non-Rigid Motions

DilationsVertical

transformationsHorizontal

transformationsEnlargement/reduction

x

y

-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

C

A

D

B(8, 4)

(2, –4)

(–1, ½)(–4, 2) E(2, 1)

(½, –1)F

Stretch/compression

x

y

-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

CA

D

B(3, 0)(0, 0)

(0, 2)

(0, 4)

Stretch/compression

x

y

-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

CA

B

(3, 0)D

(6, 0)

(0, 0)

(0, 3)

Compare ABC with DEF . The size of each side changes by a constant scale factor. The angle measures have stayed the same.

Compare ABC with ADC . The vertical distance changes by a scale factor. The horizontal distance remains the same. Two of the angles have changed measures.

Compare ABC with ABD . The horizontal distance changes by a scale factor. The vertical distance remains the same. Two of the angles have changed measures.

Common Errors/Misconceptions

• mistaking a non-rigid motion for a rigid motion and vice versa

• not recognizing that rigid motions preserve shape and size

• not recognizing that it takes only one non-rigid motion to render two figures not congruent


Instruction


© Walch Education

Example 1

Determine if the two figures below are congruent by identifying the transformations that have taken place.

A C

A'

B'

C'

B

1. Determine the lengths of the sides.

For the horizontal and vertical legs, count the number of units for the length. For the hypotenuse, use the Pythagorean Theorem, a2 + b2 = c2, for which a and b are the legs and c is the hypotenuse.

AC = 3 A′C ′ = 3

CB = 5 C ′B′ = 5

AC 2 + CB2 = AB2 A C C B A B2 2 2′ ′ + ′ ′ = ′ ′

32 + 52 = AB2 32 + 52 = A′B′2

34 = AB2 34 = A′B′2

34=AB = AB2 34=AB = A B 2′ ′

34=AB A B 34′ ′ =

The sides in the first triangle are congruent to the sides of the second triangle. Note: When taking the square root of both sides of the equation, reject the negative value since the value is a distance and distance can only be positive.

Guided Practice 1.4.2


Instruction


2. Identify the transformations that have occurred.

The orientation has changed, indicating a rotation or a reflection.

The second triangle is a mirror image of the first, but translated to the right 4 units.

The triangle has undergone rigid motions: reflection and translation.

A C

A'

B'

C'

B

Ref lection

Translation4 units right

3. State the conclusion.

The triangle has undergone two rigid motions: reflection and translation. Rigid motions preserve size and shape. The triangles are congruent.


Instruction


© Walch Education

Example 2


A

A'

C'

B'B

C


For the horizontal and vertical legs, count the number of units for the length. For the hypotenuse, use the Pythagorean Theorem, a2 + b2 = c 2, for which a and b are the legs and c is the hypotenuse.

AB = 3 A′B′ = 6

AC = 4 A′C ′ = 8

AB2 + AC 2 = CB2 A B A C C B2 2 2′ ′ + ′ ′ = ′ ′

32 + 42 = CB2 62 + 82 = C ′B′ 2

25 = CB2 100 = C ′B′ 2

25=CB = CB2 C B 100′ ′ = = C B 2′ ′

25=CB C B 100′ ′ = CB = 5 C ′B′ = 10

The sides in the first triangle are not congruent to the sides of the second triangle. They are not the same size.


Instruction



The orientation has stayed the same, indicating translation, dilation, stretching, or compression. The vertical and horizontal distances have changed. This could indicate a dilation.

3. Calculate the scale factor of the changes in the side lengths.

Divide the image side lengths by the preimage side lengths.

A B

AB

6

32

′ ′= =

A C

AC

8

42

′ ′= =

C B

CB

10

52

′ ′= =

The scale factor is constant between each pair of sides in the preimage and image. The scale factor is 2, indicating a dilation. Since the scale factor is greater than 1, this is an enlargement.

A

A'

C'

B'B

C

45

3

6

810


The triangle has undergone at least one non-rigid motion: a dilation. Specifically, the dilation is an enlargement with a scale factor of 2. The triangles are not congruent because dilation does not preserve the size of the original triangle.


Instruction


© Walch Education

Example 3


A

A'

B' C'

D'B

D

C


For the horizontal and vertical sides, count the number of units for the length.

AB = 6 A′B′ = 4.5BC = 4 B′C ′ = 4CD = 6 C ′D′ = 4.5DA = 4 D′A′ = 4

Two of the sides in the first rectangle are not congruent to two of the sides of the second rectangle. Two sides are congruent in the first and second rectangles.


Instruction



The orientation has changed, and two side lengths have changed. The change in side length indicates at least one non-rigid motion has occurred. Since not all pairs of sides have changed in length, the non-rigid motion must be a horizontal or vertical stretch or compression.

The image has been reflected since BC lies at the top of the preimage and ′ ′B C lies at the bottom of the image. Reflections are rigid motions. However, one non-rigid motion makes the figures not congruent. A non-rigid motion has occurred since not all the sides in the image are congruent to the sides in the preimage.

The vertical lengths have changed, while the horizontal lengths have remained the same. This means the transformation must be a vertical transformation.


Instruction


© Walch Education

3. Calculate the scale factor of the change in the vertical sides.

Divide the image side lengths by the preimage side lengths.

′ ′= =

A B

AB

4.5

60.75

′ ′= =

C D

CD

4.5

60.75

The vertical sides have a scale factor of 0.75. The scale factor is between 0 and 1, indicating compression. Since only the vertical sides changed, this is a vertical compression.

A

A'

B' C'

D'B

D

C

6

4

4

4.5


The vertical sides of the rectangle have undergone at least one non-rigid transformation of a vertical compression. The vertical sides have been reduced by a scale factor of 0.75. Since a non-rigid motion occurred, the figures are not congruent.


Instruction


Example 4


A

A'

B' C'

B

C


For the horizontal and vertical sides, count the number of units for the length. For the hypotenuse, use the Pythagorean Theorem, a2 + b2 = c2, for which a and b are the legs and c is the hypotenuse.

BC = 11 B′C ′ = 11

CA = 7 C ′A′ = 7

BC 2 + CA2 = AB2 ′ ′ + ′ ′ = ′ ′B C C A A B2 2 2

112 + 72 = AB2 112 + 72 = A′B′ 2

170 = AB2 170 = A′B′ 2

170=AB = AB2 170=AB = A B 2′ ′

170=AB ′ ′ =A B 170

The sides of the first triangle are congruent to the sides of the second triangle.


Instruction


© Walch Education


The orientation has changed and all side lengths have stayed the same. This indicates a reflection or a rotation. The preimage and image are not mirror images of each other. Therefore, the transformation that occurred is a rotation.


Rotations are rigid motions and rigid motions preserve size and shape. The two figures are congruent.

UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS Lesson...

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