CCSSM Stage 3 Companion Text Lesson 3-O. Warm-Up 1.Describe the translation that moves A(–3, 4) to A'(1, 3). 2.Describe the type of reflection that moves.

CCSSM Stage 3 Companion CCSSM Stage 3 Companion TextText

Lesson 3-O

Warm-Up1. Describe the translation that moves A(–3, 4) to A'(1, 3).

2. Describe the type of reflection that moves B(5, 1) to B'(–5, 1).

3. Describe the type of transformation that occurs when you multiply all x- and y-coordinates by 4.

Shifted 4 units right and 1 unit down.

Dilation with scale factor of 4.

Reflected over the y-axis.

Transformations & CongruenceTransformations & Congruence

Lesson 3-O

VocabularyCongruentTwo figures are congruent if they are the exact same shape and the exact same size.

DOG CAT

SimilarTwo figures are similar if they are the same shape but different sizes.

HOG is similar to PEN

D G

O

C T

A

H G

O

P

E

N

DOG CAT is read “triangle D-O-Gis congruent to triangle C-A-T.”

In this Explore! you will draw the image of a triangle given a transformation and then determine if the image is congruent or similar to the original triangle.

Step 1 List the four types of transformations on four separate lines of your paper.

Step 2 Each type of transformation forms an image that is either similar or congruent to the pre-image. Make a conjecture for each transformation you listed in Step 1 about whether the image will be congruent or similar to the pre-image.

Step 3 Use ABC for Steps 4 through 7:

A(2, 5), B(4, 3), and C(3, 0).

Use graph paper to make five coordinate planes (each from −10 to 10 on both axes). Graph the pre-image on the first coordinate plane.

Congruence & Similarity with TransformationsCongruence & Similarity with Transformations

ReflectionThe pre-image and its image are

congruent .

TranslationThe pre-image and its image are

congruent .

RotationThe pre-image and its image are

congruent .

DilationThe pre-image and its image are

similar.

Example 1State the type of transformation described using each transformation rule. Then state if the image will be congruent or similar to its pre-image.

a.

x – 3 means shifting left 3 units and y + 4 means shifting up 4 units.

This is a TRANSLATION.

Translations form CONGRUENT figures.

, 3, 4x y x y

Example 1 (Cont.)Describe the type of transformation described using each transformation rule. Then state if the image will be congruent or similar to its pre-image.

b.

A transformation that multiplies the coordinates of a figure by a scale factor (in this case, 4) is a DILATION.

Dilations form SIMILAR figures.

, 4 ,4x y x y

Example 2EFG was formed by a single transformation of MNP.

a. Are EFG and MNP congruent or similar?

b. Write a translation rule that maps MNP onto EFG.

a. EFG and MNP are congruent because they are the same shape and same size.

b. MNP is moved 2 units to the rightand 6 units down to make EFG.

, 2, 6x y x y

Transformation RulesTransformation RulesReflection over the y-axisThe x-coordinate changes sign.

Reflection over the x-axisThe y-coordinate changes sign.

DilationShown by multiplying the x- and y-coordinates by the scale factor.

Examples: scale factor of 5 scale factor of

, 5 ,5x y x y 1 12 2

, ,x y x y

, ,x y x y , ,x y x y

12

2,3 2,3

2, 3

)3,2(

Exit ProblemsDescribe the transformation given by each rule. State whether the transformation creates similar or congruent figures.

1. (x, y) (x, y + 8)

2. (x, y) (x, y)

3. (x, y) (x + 2, y – 1)

4. (x, y) 5 52 2,x y

Communication Prompt

How can you use transformations to show that two figures are similar?