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G.3.A Describe and perform transformations of figures in a plane using coordinate notation. Also G.3.B, G.3.C

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Explore Exploring TranslationsA translation slides all points of a figure the same distance in the same direction.

You can use tracing paper to model translating a triangle.

First, draw a triangle on lined paper. Label the vertices A, B, and C. Then draw a line segment XY. An example of what your drawing may look like is shown.

Use tracing paper to draw a copy of triangle ABC. Then copy

_ XY so that the point X is on top

of point A. Label the point made from Y as A'.

Using the same piece of tracing paper, place A' on A and draw a copy of △ABC. Label the corresponding vertices B' and C'. An example of what your drawing may look like is shown.

Use a ruler to draw line segments from each vertex of the preimage to the corresponding vertex on the new image.

Module 2 77 Lesson 1

2 . 1 TranslationsEssential Question: How do you draw the image of a figure under a translation?

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Texas Math StandardsThe student is expected to:

G.3.A

Describe and perform transformations of figures in a plane using coordinate notation. Also G.3.B, G.3.C

Mathematical Processes

G.1.F

Analyze mathematical relationships to connect and communicate mathematical ideas.

Language Objective

1.A, 3.D, 3.E, 4.G

Work with a partner to identify examples and non-examples of translations.

HARDCOVER PAGES 6170

Turn to these pages to find this lesson in the hardcover student edition.

Translations

ENGAGE Essential question: How do you draw the image of a figure under a translation?Possible answer: You can use tracing paper to slide

the figure parallel to a vector. On the coordinate

plane, you can draw the image by using coordinate

notation to calculate the coordinates of the images

of the vertices.

PREVIEW: LESSON PERFORMANCE TASKView the online Engage. Discuss the photo; ask students to describe repetitions they see in the pattern and tell whether they could create it by using copies of a smaller section. Then preview the Lesson Performance Task.

77

HARDCOVER PAGES

Turn to these pages to find this lesson in the hardcover student edition.

Resource

Locker

G.3.A Describe and perform transformations of figures in a plane using coordinate notation.

Also G.3.B, G.3.C

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Explore Exploring Translations

A translation slides all points of a figure the same distance in the same direction.

You can use tracing paper to model translating a triangle.

First, draw a triangle on lined paper. Label the

vertices A, B, and C. Then draw a line segment

XY. An example of what your drawing may look

like is shown.

Use tracing paper to draw a copy of triangle

ABC. Then copy _ XY so that the point X is on top

of point A. Label the point made from Y as A'.

Using the same piece of tracing paper, place

A' on A and draw a copy of △ABC. Label the

corresponding vertices B' and C'. An example

of what your drawing may look like is shown.

Use a ruler to draw line segments from each

vertex of the preimage to the corresponding

vertex on the new image.

Module 2

77

Lesson 1

2 . 1 Translations

Essential Question: How do you draw the image of a figure under a translation?

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77 Lesson 2 . 1

L E S S O N 2 . 1

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A

CB

A′

C′ B′

ν⇀

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Initialpoint

Terminalpoint

ν⇀

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E Measure the distances AA', BB', CC', and XY. Describe how AA', BB', and CC' compare to the length XY.

Reflect

1. Are BB', AA', and CC' parallel, perpendicular, or neither? Describe how you can check that your answer is reasonable.

2. How does the angle BAC relate to the angle B'A'C' ? Explain.

Explain 1 Translating Figures Using VectorsA vector is a quantity that has both direction and magnitude. The initial point of a vector is the starting point. The terminal point of a vector is the ending point. The vector shown may be named

‾ ⇀ EF or

⇀ v .

Translation

It is convenient to describe translations using vectors. A translation is a transformation along a vector such that the segment joining a point and its image has the same length as the vector and is parallel to the vector.

For example, BB' is a line segment that is the same length as and is parallel to vector ⇀ v .

Possible answer: BB', AA', and CC' are each 2 inches. XY is also 2 inches. The segments are

all the same length.

They are parallel. I can turn the tracing paper so that one of the lines is on one of the

parallel rules of the lined paper. All of the segments line up with the lines on the paper.

So, they are parallel.

Possible answer: The angles are congruent. Since translation is a rigid transformation, the

side lengths and angle measures remain the same in the translated figure.


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Integrate Mathematical ProcessesThis lesson provides an opportunity to address TEKS G.1.F, which calls for students to “analyze mathematical relationships to connect and communicate mathematical ideas ….” Students learn to analyze how a preimage relates both to its image and to the vector that defines the slide. Students learn the precise terms used to describe translations and the mathematical relationships represented.

EXPLORE Exploring Translations

INTEGRATE TECHNOLOGYIf you use software instead of tracing paper, begin by reviewing how to use geometry software to draw points and figures, to mark a vector, and to translate using a vector. Familiarity with those techniques will allow students to concentrate on the concepts they are exploring.

QUESTIONING STRATEGIESHow does the given vector relate to the translation of the triangle? It gives both the

distance and the direction between pairs of

corresponding points from the preimage and the

image.

PROFESSIONAL DEVELOPMENT

Translations 78


ν⇀

A

B

C

A

B

C ν⇀

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B

C

A′

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C′

ν⇀

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A′ B′

C′D′ν⇀

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You can use these facts about parallel lines to draw translations.

• Parallel lines are always the same distance apart and never intersect.

• Parallel lines have the same slope.

Example 1 Draw the image of △ABC after a translation along ⇀ v .

Draw a copy of ⇀ v with its initial point at vertex A of △ABC.

The copy must be the same length as ⇀ v , and it must be parallel

to ⇀ v . Repeat this process at vertices B and C.

Draw segments to connect the terminal points of the vectors. Label the points A', B', and C'. △A'B'C' is the image of △ABC.

Draw a vector from the vertex A that is the same length as and vector

⇀ v . The terminal point A' will

be units up and 3 units .

Draw three more vectors that are parallel from , ,

and with terminal points B', C', and D'.

Draw segments connecting A', B', C', and D' to

form .

5

parallel to

left

B C

D

quadrilateral A'B'C'D'


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COLLABORATIVE LEARNING

Peer-to-Peer ActivityHave students work in pairs. Instruct one student in each pair to give an example of a translation that would move a given figure from Quadrant I to Quadrant III, and write steps to show what was done. The partner then checks this work. Then they switch roles and repeat, using a translation between two different quadrants.

EXPLAIN 1 Translating Figures Using Vectors

AVOID COMMON ERRORSSome students may assume that they should slide the figure along the vector. Compare a vector to aspects of a map such as scale or location of north and south, since the vector gives information about the direction and distance of the move.

COOPERATIVE LEARNINGHave students pair up to check each other’s drawings of the vectors and translated figures. Remind students to check the labels on their partners’ drawings.

79 Lesson 2 . 1


A B

C

ν⇀

A′B′

C′

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Reflect

3. How is drawing an image of quadrilateral ABCD like drawing an image of △ABC? How is it different?

Your Turn

4. Draw the image of △ABC after a translation along ⇀ v .

Explain 2 Drawing Translations on a Coordinate PlaneA vector can also be named using component form, ⟨a, b⟩, which specifies the horizontal change a and the vertical change b from the initial point to the terminal point. The component form for

_ PQ is ⟨5, 3⟩.

You can use the component form of the vector to draw coordinates for a new image on a coordinate plane. By using this vector to move a figure, you are moving the x-coordinate 5 units to the right. So, the new x-coordinate would be 5 greater than the x-coordinate in the preimage. Using this vector you are also moving the y-coordinate up 3 units. So, the new y-coordinate would be 3 greater than the y-coordinate in the preimage.

Rules for Translations on a Coordinate Plane

Translation a units to the right (x, y) → (x + a, y) Translation a units to the left (x, y) → (x – a, y) Translation b units up (x, y) → (x, y + b) Translation b units down (x, y) → (x, y – b)

So, when you move an image to the right a units and up b units, you use the rule (x, y) → (x + a, y + b) which is the same as moving the image along vector ⟨a, b⟩.

Possible answer: The steps to drawing an image of the quadrilateral are the same as

drawing an image of the triangle. It is different because there is an extra vector when

drawing the quadrilateral. Also, you must be careful to connect the vertices in the same

order as the original shape.


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DIFFERENTIATE INSTRUCTION

Critical ThinkingRemind students that they use the absolute value of the difference of coordinates when finding a vertical or horizontal distance. Ask, why don’t we use the absolute value when finding the component form of a vector? Direction is an important

part of a vector, whether positive or negative. Using the absolute value would

make all components positive.

EXPLAIN 2 Drawing Translations on a Coordinate Plane

INTEGRATE MATHEMATICAL PROCESSESFocus on Math ConnectionsRelate the concept of slope, the ratio of the vertical change to the horizontal change between two points, to the component form of the vector.

QUESTIONING STRATEGIESWhat is the slope of the line that contains the vector ⧼a, b⧽? b _ a

Describe how two vectors could have the same slope but not be the same vector. They could

have different magnitudes.

AVOID COMMON ERRORSSome students may try to position the preimage at the beginning of the vector rather than the reverse. Help them position the vector at the vertices of the preimage.

AVOID COMMON ERRORSStudents sometimes confuse translation with transformation. Have them draw a Venn diagram to make clear that translation, or slide, is one kind of transformation.

CONNECT VOCABULARY Relate the word translation in geometry to translating languages. For example, when we translate a sentence from English to Spanish, we want to keep the same meaning, from each word, for the translation to be accurate. When we translate an image, every point and line in that image is moved in the same direction and the same distance.

Translations 80


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Example 2 Calculate the vertices of the image figure. Graph the preimage and the image.

Preimage coordinates: (−2, 1) , (−3, -2) , and (−1, −2) . Vector: ⟨4, 6⟩

Predict which quadrant the new image will be drawn in: 1 st quadrant.

Use a table to record the new coordinates.

Preimage coordinates (x, y)

Image (x + 4, y + 6)

(–2, 1) (2, 7)

(–3, –2) (1, 4)

(–1, –2) (3, 4)

Then use the preimage coordinates to draw the preimage, and use the image coordinates to draw the new image.

Preimage coordinates: A (3, 0) , B (2, −2) , and C (4, −2) . Vector ⟨−2, 3⟩

Prediction: The image will be in Quadrant .


Image

(x − , y + )

(3, 0) ( , )

(2, −2) ( , )

(4, −2) ( , )

Step 1: Use vector components to write the transformation rule. Step 2: Write the

coordinates of the preimage. Step 3: Apply the

transformation rule to each coordinate of the preimage, which results in the coordinate of the image.

2 3

1

3

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1

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0

2


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LANGUAGE SUPPORT

Connect VocabularyHelp students understand the difference between the initial, or beginning, point of a vector and the terminal, or ending, point. Discuss the effects of reversing the points. When talking about the component form of a vector, emphasize the importance of listing the horizontal (left and right) change first, and then the vertical (up and down) change. Point out how reversing the two numbers changes the definition of the vector and, therefore, the location of the image it defines.

81 Lesson 2 . 1


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Your Turn

Draw the preimage and image of each triangle under a translation along ⟨−4, 1⟩.

5. Triangle with coordinates: A (2, 4) , B (1, 2) , C (4, 2) .

6. Triangle with coordinates: P (2, –1) , Q (2, –3) , R (4, –3) .

Explain 3 Specifying Translation VectorsYou may be asked to specify a translation that carries a given figure onto another figure. You can do this by drawing the translation vector and then writing it in component form.

Example 3 Specify the component form of the vector that maps △ABC to △A'B'C'.

Draw the vector ⇀ v from a vertex of △ABC to its image in △A'B'C'.

Determine the components of ⇀ v .

The horizontal change from the initial point (−4, 1) to the terminal point (1, −3) is 1 − (−4) = 5.

The vertical change from the initial point (−4, 1) to the terminal point (1, −3) is −3 − 1 = −4

Write the vector in component form.

⇀ v = ⟨5, -4⟩





EXPLAIN 3 Specifying Translation Vectors

AVOID COMMON ERRORSSome students may make an error when they subtract negative coordinates. Review the rules for subtracting integers, as needed.

QUESTIONING STRATEGIESIn writing the component form of the vector, how would you indicate that there is no

vertical change in the position of the figure? Use the

number 0 to indicate no change.

INTEGRATE MATHEMATICAL PROCESSESCognitive StrategiesDiscuss what happens when a point moves from one end of a vector to the other end, and how this relates to the translation of the vertices in the image figure.

INTEGRATE MATHEMATICAL PROCESSESMultiple RepresentationsAsk students which method for specifying the translation of a figure they find most helpful in visualizing the image. Have them discuss the advantages of each kind of description.

Translations 82


A

C

B

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B Draw the vector ⇀ v from a vertex of △ABC to its image

in △A'B'C'.

Determine the components of ⇀ v .

The horizontal change from the initial point (–3, 1) to the terminal point (2, 4) is – = .

The vertical change from the initial point to the terminal point is – =

Write the vector in component form. ⇀ v = ( , )

Reflect

7. What is the component form of a vector that translates figures horizontally? Explain.

Your Turn

8. In Example 3A, suppose △A'B'C' is the preimage and △ABC is the image after translation. What is the component form of the translation vector in this case? How is this vector related to the vector you wrote in Example 3A?

Elaborate

9. How are translations along the vectors ⟨a, −b⟩ and ⟨−a, b⟩ similar and how are they different?

10. A translation along the vector ⟨−2, 7⟩ maps point P to point Q. The coordinates of point Q are (4, −1) . What are the coordinates of point P? Explain your reasoning.

11. A translation along the vector ⟨a, b⟩ maps points in Quadrant I to points in Quadrant III. What can you conclude about a and b? Justify your response.

12. Essential Question Check-In How does translating a figure using the formal definition of a translation compare to the previous method of translating a figure?

3

2

4

-3

1

5

3

5

The vector has the form ⟨a, 0⟩. There is no change in the vertical direction, so the value of

b is 0.

⟨-5, 4⟩; the components are the opposites of the components of the vector in Example 3A.

(6, −8); Solving equations x − 2 = 4 and y + 7 = −1 shows that x = 6 and y = −8.

Both a and b are negative. Points in Quadrant I have positive x- and y-coordinates and

points in Quadrant III have negative x- and y-coordinates. This means the horizontal and

vertical changes are both negative. So a is negative and b is negative.

Possible answer: Rather than sliding a figure left or right and then up or down to translate

it, you slide it parallel to a given vector a distance equal to the length of the vector.

They move the points the same distance, but in opposite directions.




ELABORATE QUESTIONING STRATEGIES

What can you say about the length of the line segments drawn from each vertex to its image?

Each is equal to the magnitude of the vector used to

draw the translation.

SUMMARIZE THE LESSONGive at least three different ways to describe a translation. Sample answer: A translation is

(1) a slide that moves the figure so many units

horizontally and so many vertically;

(2) moving all points of the figure the same distance

in the same direction;

(3) drawing an image by sliding a preimage parallel

to a given vector a distance equal to the length of

the vector;

(4) using a pair of coordinates to find the

coordinates of the images of the vertices.

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C

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BC

ν⇀

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B′C′

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Evaluate: Homework and Practice

Draw the image of △ABC after a translation along ⇀ v .

1. 2.

3. 4.

5. Line segment _ XY was used to draw a copy of △ABC.

_ XY is 3.5 centimeters long. What

is the length of AA' + BB' + CC'?

10.5 cm





Exercise Depth of Knowledge (D.O.K.) Mathematical Processes

1–4 2 Skills/Concepts 1.E Create and use representations

5 1 Recall of Information 1.F Analyze relationships


9 1 Recall of Information 1.D Multiple representations

10 1 Recall of Information 1.F Analyze relationships

11 2 Skills/Concepts 1.C Select tools

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

Example 1Translating Figures Using Vectors

Exercises 1-4

ExploreExploring Translations

Exercise 5

Example 2Drawing Translations on a Coordinate Plane

Exercises 6–8

Example 3Specifying Translation Vectors

Exercises 13–16

INTEGRATE MATHEMATICAL PROCESSESFocus on PatternsEncourage students to predict, before constructing the translated image, where the image will lie, using the distance and direction given by the translation vector.

INTEGRATE MATHEMATICAL PROCESSESFocus on CommunicationProvide students with graphic examples of translated images (preimage and image) and preimages that may have been transformed but are not translations. Have students sort the pictures into translations and non-translations, discuss why they are or are not translations, and write the justifications on note cards. Provide a list of key terms for students to use in their explanations.

Translations 84


y

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B

A C

B′

A′ C′

y

0x

4

-2

2

-22 4

XZ

YX′

Z′

Y′

y

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-2

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P R

P′ R′

Q′

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Draw the preimage and image of each triangle under the given translation.

6. Triangle: A (-3, -1) ; B (-2, 2) ; C (0, -1) ; Vector: ⟨3, 2⟩

7. Triangle: P (1, -3) ; Q (3, -1) ; R (4, -3) ; Vector: ⟨-1, 3⟩

8. Triangle: X (0, 3) ; Y (−1, 1) ; Z (–3, 4) ; Vector: ⟨4, -2⟩



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Exercise Depth of Knowledge (D.O.K.) Mathematical Processes

12 3 Strategic Thinking 1.C Select tools


17–18 3 Strategic Thinking 1.G Explain and justify arguments

19 2 Skills/Concepts 1.G Explain and justify arguments

20 3 Strategic Thinking 1.G Explain and justify arguments

21–22 2 Skills/Concepts 1.G Explain and justify arguments

23 3 Strategic Thinking 1.G Explain and justify arguments

INTEGRATE MATHEMATICAL PROCESSESModelingHave students draw a vector on graph paper and then use it as the hypotenuse of a right triangle. Discuss how the legs of the triangle show the vertical and horizontal components.

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9. Fill in the table to find the coordinates of the image under the transformation ⟨6, -11⟩.


Image

(x + , y − )

(3, 1) ( , ) (2, −3)

( , ) (4, −3) ( , )

10. Name the vector and write it in component form.

11. Match each set of coordinates for a preimage with the coordinates of its image after applying the vector ⟨3, -8⟩. Indicate a match by writing a letter for a preimage on the line in front of the corresponding image.

A. (1, 1) ; (10, 1) ; (6, 5) (6, -10) ; (6, -4) ; (9, -3)

B. (0, 0) ; (3, 8) ; (4, 0) ; (7, 8) (1, -6) ; (5, -6) ; (-1, -8) ; (7, -8)

C. (3, -2) ; (3, 4) ; (6, 5) (4, -7) ; (13, -7) ; (9, -3)

D. (-2, 2) ; (2, 2) ; (-4, 0) ; (4, 0) (3, -8) ; (6, 0) ; (7, -8) ; (10, 0)

12. Persevere in Problem Solving Emma and Tony are playing a game. Each draws a triangle on a coordinate grid. For each turn, Emma chooses either the horizontal or vertical value for a vector in component form. Tony chooses the other value, alternating each turn. They each have to draw a new image of their triangle using the vector with the components they chose and using the image from the prior turn as the preimage. Whoever has drawn an image in each of the four quadrants first wins the game.

Emma’s initial triangle has the coordinates (-3, 0) , (- 4, -2) , (-2, -2) and Tony’s initial triangle has the coordinates (2, 4) , (2, 2) , (4, 3) . On the first turn the vector ⟨6, -5⟩ is used and on the second turn the vector ⟨-10, 8⟩ is used. What quadrant does Emma need to translate her triangle to in order to win? What quadrant does Tony need to translate his triangle to in order to win?

C

A

D

B

Emma: Quadrant I; Tony: Quadrant III

6 11

9

8

-10

-14

10 -14

⇀ GH , ⟨5, -2⟩




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AVOID COMMON ERRORSWhen writing the component form of a given vector, some students may reverse the initial point and the terminal point. Suggest that they circle the arrow that shows the terminal point before writing the components.

Translations 86


y

0x

4

-2-2

A

C

B

A′

C′

B′

y

0x

4

-2-2

AB

A′

C′

B′

C

2

4

y

0-4

4

x

-2

2

2

CD

D′

A B

C′

A′ B′

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x

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2

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A′

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Specify the component form of the vector that maps each figure to its image.

13.

14.

15.

16.

⟨4, -3⟩

⟨2, 3⟩

⟨2, -5⟩

⟨5, 0⟩

Answers may vary. Sample: He drew vectors from A to A' and from B to B' that were not parallel to or the same length as

⇀ v . The correct vectors should each point 3 units right and

3 units down.

He should have drawn the vector from F to F’ or from E to E’. The correct component form of the vector is ⟨5, 1⟩.




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CA′

B′

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17. Explain the Error Andrew is using vector ⇀ v to draw a copy of △ABC. Explain his error.

18. Explain the Error Marcus was asked to identify the vector that maps ΔDEF to ΔD'E'F'. He drew a vector as shown and determined that the component form of the vector is ⟨3, 1⟩. Explain his error.

19. Algebra A cartographer is making a city map. Line m represents Murphy Street. The cartographer translates points on line m along the vector ⟨2, -2⟩ to draw Nolan Street. Draw the line for Nolan Street on the coordinate plane and write its equation. What is the image of the point (0, 3) in this situation?

H.O.T. Focus on Higher Order Thinking

20. Represent Real-World Problems A builder is trying to level out some ground with a front-end loader. He picks up some excess dirt at (9, 16) and then maneuvers through the job site along the vectors ⟨-6, 0⟩, ⟨2, 5⟩, ⟨8, 10⟩ to get to the spot to unload the dirt. Find the coordinates of the unloading point. Find a single vector from the loading point to the unloading point.

Answers may vary. Sample: He drew vectors from A to A' and from B to B' that were not parallel to or the same length as

⇀ v . The correct vectors should each point 3 units right and

3 units down.

He should have drawn the vector from F to F’ or from E to E’. The correct component form of the vector is ⟨5, 1⟩.

y = 2x - 3; (2, 1)

(13, 31) ; ⟨4, 15⟩




GE_MTXESE353886_U1M02L1.indd 88 21/01/15 12:26 PM

Translations 88


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21. Look for a Pattern A checker player’s piece begins at K and, through a series of moves, lands on L. What translation vector represents the path from K to L?

22. Represent Real-World Problems A group of hikers walks 2 miles east and then 1 mile north. After taking a break, they then hike 4 miles east to their final destination. What vector describes their hike from their starting position to their final destination? Let 1 unit represent 1 mile.

23. Communicate Mathematical Ideas In a quilt pattern, a polygon with vertices (-4, -2) , (-3, -1) , (-2, -2) , and (-3, -3) is translated repeatedly along the vector ⟨2, 2⟩. What are the coordinates of the third polygon in the pattern? Explain how you solved the problem.

⟨6, 1⟩

⟨4, -6⟩

(0, 2) ; (1, 3) ; (2, 2) ; (1, 1) ; Possible answer: I used a table to find the coordinates of the second polygon. Then I made a new table, using the coordinates from the second polygon to find the coordinates of the third polygon.




JOURNALHave students describe what a vector is and how it is used to define a translation.

89 Lesson 2 . 1


Image A

Image B

Image C

Image D

Image A

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Lesson Performance Task

A contractor is designing a pattern for tiles in an entryway, using a sun design called Image A for the center of the space. The contractor wants to duplicate this design three times, labeled Image B, Image C, and Image D, above Image A so that they do not overlap. Identify the three vectors, labeled

⇀ m ,

⇀ n , and

⇀ p that could be used to draw the design, and write them in

component form. Draw the images on grid paper using the vectors you wrote.

Drawings and vectors will vary. Possible drawing and vectors are shown.

⇀ m moves Image A to Image B.

⇀ m = ⟨-8, 6⟩

⇀ n moves Image A to Image C.

⇀ n = ⟨0, 8⟩

⇀ p moves Image A to Image D.

⇀ p = ⟨8, 6⟩



GE_MTXESE353886_U1M02L1 90 2/27/14 7:55 AM

CONNECT VOCABULARY Ask students to describe any similarities between the meaning of translation as it is used in this lesson and the word’s meaning when it is used to describe the process of converting words from one language to another. Sample answer: Each transforms

something into something else. In this lesson, a

translation transforms a point to a point in another

position. A language translation transforms a word

in one language into an equivalent word in another

language.

QUESTIONING STRATEGIESSuppose

‾→ m = (3, –5) translates P (4, –2) to Q.

How do you find the coordinates of Q? Add 3

to 4 for the x-coordinate and add –5 to –2 for the

y-coordinate.

What is a vector ‾→ n that will translate Q to P?

Why? ‾→ n = (–3, 5) ; each component is the

opposite of the component in the translation vector.

EXTENSION ACTIVITY

Provide students with grid paper. Have each student design and color a geometric shape. The design should be no larger than a 6 × 6 portion of the grid. The student should write three or more vectors indicating where additional designs like theirs are to be drawn, making sure that translated designs do not overlap the original or each other. Students then exchange grids with a partner and draw the translated designs indicated by their partner’s vectors.

Scoring Rubric2 points: Student correctly solves the problem and explains his/her reasoning.1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning.0 points: Student does not demonstrate understanding of the problem.

Translations 90


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CorrectionKey=TX-B LESSON 2 . 1 CorrectionKey=TX-A DO NOT … · 2015-09-13 · Label the vertices A, B, and C. Then draw a line segment XY. An example of what your drawing may look