Coordinate 5 Geometry and Transformationsclassroom.dickinsonisd.org/users/0315/docs/cb_sbo_math_miu_l5_u5... · Unit5 349 Coordinate ... Embedded Assessment 1 Transformations p. 369

© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

5Unit

349

CoordinateGeometry and Transformations

Essential Questions

What connections exist between transformations and dilations and congruence and similarity?

How are transformations and tessellations used in real world settings?

Unit OverviewIn this unit you will investigation the derivations of formulas for the area of plane fi gures and study transformations and tessellations of fi gures. You will be introduced to a non-Euclidean geometry, vectors, and matrices.

Academic VocabularyAdd these words the academic vocabulary portion of your math notebook.

rotation vector dilation

tessellation matrix

This unit has three Embedded Assessments. Completing these embedded assessments will allow you to demonstrate your understanding of transformation and dilations on a coordinate plane and your understanding of the area of plane fi gures. You will also show your knowledge of vectors and matrices.

Embedded Assessment 1

Transformations p. 369


Tessellations p. 385


Matrices, Transformations, and Vectors p.403

EMBEDDED ASSESSMENTS

??

??

349-350_SB_Geom_U5_UO_SE.indd 349349-350_SB_Geom_U5_UO_SE.indd 349 1/21/10 11:20:05 AM1/21/10 11:20:05 AM

© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

350 SpringBoard® Mathematics with Meaning™ Geometry

Write your answers on notebook paper.

1. Tell whether the following statements are sometimes, always, or never true.a. A rhombus has 4 right anglesb. Opposite sides of a square are parallel and

congruentc. A trapezoid is a parallelogramd. Adjacent angles of a rectangle are

complementary.

2. Quadrilateral ABCD has vertices (–2, 6), (2, 3), (2, –2), and (–2, 1).a. Graph ABCD.b. What special quadrilateral is ABCD? Justify

your answer.

3. Triangle PQR has vertices (1, 2), (4, 9) and (7, 2). Draw a triangle in the 3rd quadrant that is congruent to �PQR.

4. What is the diff erence in the lengths of the hypotenuses of the right triangles shown below?Note: Triangles not drawn to scale

45°

1 unit1 unit

30°

5. Solve. a. B =

2 __ 5 cd for c.

b. x = 1 __ 2 yz + y for z.

6. Tell whether each of the following statements is true or false. If the statement is false explain why it is false.In a 30/60/90 degree triangle:a. If the shorter leg is 3 units then the

hypotenuse measures 6 units.b. If the shorter leg measures 4 units, then the

longer leg measures 2 √__

3 units.c. Th e longest leg is always √

__ 3 times longer

than the hypotenuse.In a 45/45/90 degree triangle: d. the length of the hypotenuse is 2 times the

length of either leg.

7. Find each of the following ratios using the triangle pictured.a. cos Rb. sin Rc. tan R

P

QR

UNIT 5

Getting Ready

349-350_SB_Geom_U5_UO_SE.indd 350349-350_SB_Geom_U5_UO_SE.indd 350 1/21/10 11:20:13 AM1/21/10 11:20:13 AM

© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

ACTIVITY

Unit 5 • Coordinate Geometry and Transformations 351

5.1Transformations on the Coordinate PlaneHalftime SaluteSUGGESTED LEARNING STRATEGIES: Questioning the Text, Shared Reading, Visualization

To boost school spirit and get students excited about geometry, the marching band at MIU High School is planning a halft ime show that features formations in the shape of special quadrilaterals. Th e band director, Mr. Scott, asks band members who are also on the Math Team to help him plan the show.

Th e marching band decides to use a Cartesian plane to represent the football fi eld. Th e diagram below shows the MIU HS football fi eld in the Cartesian plane. Th e band members locate the origin (0,0) at the intersection of the 50-yard line and the center line on the football fi eld. Each unit in the Cartesian plane is equal to 1 yard on the fi eld.

An actual football fi eld is 100 yards long and 53.3 yards wide. Positions on the fi eld are described relative to the middle of the fi eld where the 50-yard line and the horizontal center line intersect one another. Positions to the left of the 50-yard line have negative coordinates and positions to the right of the 50-yard line have positive coordinates. Th e center line divides the football fi eld into “Home” and “Away” sides.

20A

20H

10A

10H

10L 20L 30L 40L 50 40R 30R 20R 10R

End

Zone

10L 20L 30L 40L 50 40R 30R 20R 10R

20A

20H

10A

10H

End Zone-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50

2525

2020

1515

1010

5

-5-5

-10-10

-15-15

-20-20

-25-25

25

20

15

10

5

-5

-10

-15

-20

-25

Away

Home

T

S

351-362_SB_Geom_5-1_SE.indd 351351-362_SB_Geom_5-1_SE.indd 351 1/21/10 12:31:18 PM1/21/10 12:31:18 PM

© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes

ACTIVITY 5.1continued

Transformations on the Coordinate PlaneHalftime SaluteHalftime Salute

1. Band member Zack Cefone starts the fi rst song, “Going the Distance,” at the coordinates (–20, 0). Plot Zack’s location on the diagram of the fi eld on the previous page. How would you explain to Zack where he starts the halft ime show in terms of his position on the football fi eld?

2. Th e band director asks Sue to start the halft ime show on the 50-yard line, 15 yards above the center line. Plot Sue’s location on the diagram of the fi eld on the previous page. What are Sue’s starting position coordinates?

3. Zack and Sue are at the opposite ends of a line of 11 band members. Draw the line segment between Zack and Sue on the diagram on the previous page. How far apart are Zack and Sue? Show work to justify your answer.

4. When the music begins, Zack and Sue’s line is going to march to a new location on the fi eld. Zack marches to (–30, –15). Sue marches to (–10, 0). Draw Zack and Sue’s line in its new location on the diagram of the football fi eld.

a. How far did Zack and Sue travel to get to their new locations?

b. Did the length of Zack and Sue’s line change? Explain your reasoning.

5. Th e Math Team members tell Mr. Scott that the movement described in Item 4 is called a translation. Write a defi nition for the term translation in your own words.

6. Th e transformation notation for a translation is:

(x, y) → (x + a, y + b)

where a represents the change in the x-coordinates and b represents the change in the y-coordinates. Use transformation notation to represent the translation in Item 4.

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Close Reading, Interactive Word Wall, Vocabulary Organizer, Think/Pair/Share, Group Presentation, Create Representations, Quickwrite, Self/Peer Revision


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes

7. Translate Zack and Sue’s line from Item 4(a) so that Sue ends up at the center of the football fi eld.

a. What are Zack and Sue’s coordinates now?

b. How far did Sue move?

c. How far did Zack move?

d. Did all of the band members on Zack and Sue’s line move the same distance as Zack and Sue? Explain your reasoning.

e. Use transformation notation to express this translation.

8. For the next formation, Sue marches in place at the origin and Zack moves parallel to the center line, stopping on the 30-yard line to the right of the 50-yard line. Th e rest of the band line also moves parallel to the center line, stopping on the same yard line on the opposite side of the fi eld as their starting points.

a. What are Zack’s coordinates at his new location?

b. Draw Zack and Sue’s line in its new location. How far did Zack travel to get to his new location? Show work to justify your answer.

c. Did all of the band members on Zack and Sue’s line move the same distance as Zack and Sue? Explain your reasoning.

9. Th e type of movement described in Item 8 is called a refl ection. Th e line was refl ected over the 50-yard line (y-axis). Defi ne the term refl ection in your own words.

10. Consider the coordinates for each band member on the line in Item 8.

a. How did the x-coordinate change for each member?

b. How did the y-coordinate change for each member?

c. Express this refl ection in transformation notation.

SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Vocabulary Organizer, Think/Pair/Share, Group Presentation, Create Representations, Look for a Pattern, Quickwrite, Self/Peer Revision

Transformations on the Coordinate PlaneHalftime SaluteHalftime Salute



© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My NotesMy Notes

Transformations on the Coordinate PlaneHalftime Salute Halftime Salute


11. To get to the line’s fi nal formation, refl ect Zack and Sue’s line across the right side 30-yard line.

a. Write instructions to tell the band line members how they should move to get to their new positions.

b. What are Zack and Sue’s coordinates now?

c. How far did Zack move?

d. How far did Sue move?

12. Claire Anett is the band member in the middle of Zack and Sue’s line. Recall that Zack started at (-20, 0) and Sue started at (0, 15).

a. What were Claire’s coordinates when Zack and Sue’s line started the song? Explain your answer.

b. What are Claire’s coordinates once the band line is in its fi nal location? Show work to justify your answer.

c. How many total yards did Claire travel during this song? Show work to justify your answer.

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Close Reading, Think/Pair/Share, Group Presentation, Identify a Subtask, Quickwrite, Self/Peer Revision


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


Transformations on the Coordinate Plane Halftime SaluteHalftime Salute

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Marking the Text, Think/Pair/Share, Group Presentation, Identify a Subtask, Quickwrite, Self/Peer Revision

At the beginning of the band’s second song, “Th e Quadrilateral Quickstep,” Zack and Sue’s line is part of a quadrilateral formation with other band members. Zack’s position is (20, -15) and Sue’s position is at (40, 0). Th e other vertices of this new quadrilateral are Tom Pratt at (40, 15) and Patty Drum at (0, -15).

20A

20H

10A

10H

10L 20L 30L 40L 50 40R 30R 20R 10R

End

Zone

10L 20L 30L 40L 50 40R 30R 20R 10R

20A

20H

10A

10H

End Zone-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50

2525

2020

1515

1010

5

-5-5

-10-10

-15-15

-20-20

-25-25

25

20

15

10

5

-5

-10

-15

-20

-25

Away

Home

T

S

13. Plot the coordinates of the vertices of the quadrilateral and draw the band formation on the diagram above. Label the vertices using the fi rst letter of each player’s name.

a. Describe Zack, Sue, Tom, and Patty’s positions on the football fi eld.

b. What is the best name for Quad ZSTP? Show the work that supports your answer.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes

Transformations on the Coordinate Plane ACTIVITY 5.1continued Halftime SaluteHalftime Salute

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Think/Pair/Share, Identify a Subtask, Self/Peer Revision

14. For the next formation, the bases of the trapezoid ( ___

PT and __

ZS ) march toward each other and form the median of Quad PTSZ.

a. What are the coordinates of the midpoints of the legs of this trapezoid?

b. Plot the midpoints and draw the new formation made by the band. How long is the median made by the band members?

c. Show that the length of the median is half the sum of the length of the bases.

20A

20H

10A

10H

10L 20L 30L 40L 50 40R 30R 20R 10R

End

Zone

10L 20L 30L 40L 50 40R 30R 20R 10R

20A

20H

10A

10H

End Zone-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50

25

20

15

10

5

-5

-10

-15

-20

-25

25

20

15

10

5

-5

-10

-15

-20

-25

Away

Home

T

S

P

S

Z

T20A

20H

10A

10H

10L 20L 30L 40L 50 40R 30R 20R 10R

End

Zone

10L 20L 30L 40L 50 40R 30R 20R 10R

20A

20H

10A

10H

End Zone-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50

25

20

15

10

5

-5

-10

-15

-20

-25

25

20

15

10

5

-5

-10

-15

-20

-25

Away

Home

T

S

P

S

Z

T

Recall that the median of a trapezoid is the segment that joins the midpoints of the legs of the trapezoid.

MATH TERMS


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes



SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Marking the Text, Think/Pair/Share, Identify a Subtask, Self/Peer Revision

20A

20H

10A

10H

10L 20L 30L 40L 50 40R 30R 20R 10R

End

Zone

10L 20L 30L 40L 50 40R 30R 20R 10R

20A

20H

10A

10H

End Zone-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50

25

20

15

10

5

-5

-10

-15

-20

-25

25

20

15

10

5

-5

-10

-15

-20

-25

Away

Home

T

S

ZP

T

S

15. Th e band is going to form a parallelogram at the end of “Th e Quadri-lateral Quickstep.” Zack, Sue, Tom, and Patty are again at the vertices as shown above. Th ey march to new positions Z(20, −25), S(50, 0), T(30, 20) and P(0, −5) to form the parallelogram. Plot and label the vertices of the parallelogram on the diagram above.

a. Verify that Quad ZSTP is a parallelogram. Show your work.

b. Th e band members located along __

ZP moved to their new formation by rotating about the midpoint of the line. Compare the new position of

__ ZP to its old position. Th rough how many

degrees about its midpoint did __

ZP rotate as the band moved into formation?

ACADEMIC VOCABULARY

A rotation is a transformation in which each point of the pre-image travels clockwise or counterclockwise around a fi xed point a certain number of degrees.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Think/Pair/Share, Group Presentation, Create Representations, Quickwrite, Self/Peer Revision

20A

20H

10A

10H

10L 20L 30L 40L 50 40R 30R 20R 10R

End

Zone

10L 20L 30L 40L 50 40R 30R 20R 10R

20A

20H

10A

10H

End Zone-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50

25

20

15

10

5

-5

-10

-15

-20

-25

25

20

15

10

5

-5

-10

-15

-20

-25

Away

Home

T

S

Z

P

16. During this song, the drum major moves to the center of the parallelogram.

a. Draw in the diagonals of the parallelogram above and fi nd the coordinates of the midpoint of each diagonal.

b. Which property of a parallelogram does this illustrate? Explain your reasoning.

c. Describe the drum major’s position on the football fi eld in the diagram above.

17. Mr. Scott notices that the fi nal formation looks like a rectangle, but the Math Team disagrees. Who is correct? Show work to justify your answer.

The center of a quadrilateral is the point of intersection of its diagonals.

MATH TERMS


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes



SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Summarize/Paraphrase/Retell, Visualization, Think/Pair/Share, Create Representations, Identify a Subtask, Quickwrite, Self/Peer Revision

18. For the third song in the halft ime show, the band is marching to “Rhombi Are Forever.” As the song begins, the band collapses their parallelogram to form a line along the center line to the right side of the 50-yard line, as shown by the dotted line below.

20A

20H

10A

10H

10L 20L 30L 40L 50 40R 30R 20R 10R

End

Zone

10L 20L 30L 40L 50 40R 30R 20R 10R

20A

20H

10A

10H

End Zone-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50

25

20

15

10

5

-5

-10

-15

-20

-25

25

20

15

10

5

-5

-10

-15

-20

-25

Away

Home

a. Patty is on the left end at (0, 0) and Sue is on the right end of the line. What are Sue’s coordinates?

b. Tom and Zack are standing back-to-back at the midpoint of this line. What are their coordinates?

c. Th e band moves into formation as the song begins. Zack marches 5 yards toward the Home side along the 25-yard line and Tom marches 5 yards toward the Away side along the 25-yard line. Th e ends of the band line march in place. Th e rest of the band line splits with every other band member, either marching toward the Home side with Zack or toward the Away side with Tom. Th e band forms a new quadrilateral. Draw the band’s new formation on the diagram above. What is the best name for this quadrilateral? Explain your answer.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Think/Pair/Share, Group Presentation, Create Representations, Quickwrite, Self/Peer Revision

19. Mr. Scott wants Tom and Zack to continue leading their band lines above and below the center line to create a larger quadrilateral. He wants Sue and Patty to continue marching in place.

20A

20H

10A

10H

10L 20L 30L 40L 50 40R 30R 20R 10R

End

Zone

10L 20L 30L 40L 50 40R 30R 20R 10R

20A

20H

10A

10H

End Zone-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50

2525

2020

1515

1010

5

-5-5

-10-10

-15-15

-20-20

-25-25

25

20

15

10

5

-5

-10

-15

-20

-25

Away

Home

a. Use the diagram above to draw the rhombus formed by the band when Tom is 10 yards above and Zack is 10 yards below thecenter line. What happens if the entire formation is refl ected over the x-axis?

b. Where should Tom and Zack stop so their quadrilateral is a square? Draw your solution on the diagram above and explain your reasoning.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes

ACTIVITY 5.1continuedHalftime SaluteHalftime Salute

SUGGESTED LEARNING STRATEGIES: Marking the Text, Visualization, Think/Pair/Share, Group Presentation, Create Representations, Quickwrite, Self/Peer Revision

20. For the fi nal song, “Regular and Righteous,” Zack, Sue, Tom, and Patty move from their previous positions to the vertices of one fi nal quadrilateral with the rest of the band along the fi gure’s sides. Th eir fi nal coordinates are: Z(45, −15), S(40, 20), T (5, 15) and P(10, −20).

20A

20H

10A

10H

10L 20L 30L 40L 50 40R 30R 20R 10R

End

Zone

10L 20L 30L 40L 50 40R 30R 20R 10R

20A

20H

10A

10H

End Zone-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50

2525

2020

1515

1010

5

-5-5

-10-10

-15-15

-20-20

-25-25

25

20

15

10

5

-5

-10

-15

-20

-25

Away

Home

T

S

a. Sketch the square that you drew on the previous page and draw the fi nal quadrilateral. What transformation(s) did the band use to move into their fi nal formation: refl ection, rotation ortranslation? Explain your reasoning.

b. Mr. Scott wants the fi nal formation to be centered on the fi eld. Which transformation(s) will accomplish this? Where should the vertices be located?

c. Consider the transformations for Quad TPZS in Parts a and b. What happens to the formation if the order is switched? Where would the vertices be located?

21. How will a geometric fi gure, called the pre-image, compare to its image, once it is translated, refl ected, or rotated? Explain your answer.

Transformations on the Coordinate Plane


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.



CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

1. Given Quad TRAN with vertices T(1, 1), R(2, 4), A(6, 4) and N(4, −1). Perform each of the following transformations on Quad TRAN and list the coordinates of the vertices of the image. What is the best name for the transformation?

a. (x, y) → (x - 4, y)

b. (x, y) → (x, −y)

c. (x, y) → (1 − x, y)

2. Let __

AB be the pre-image with coordinatesA(2, 3) and B(−2, 0).

a. Find the slope and length of __

AB .

b. Perform the transformation(x, y) → (x − 3, y + 1) on

__ AB . How do the

slope and length of the image compare to the slope and length of the pre-image?

c. Perform the transformation(x, y) → (−x, y) on

__ AB . How do the slope

and length of the image compare to the slope and length of the pre-image?

3. Given the line with equation y = 2 __ 3 x + 1.

Write the equation of the image of the line under the given transformation.

a. (x, y) → (x, y − 3)

b. (x, y) → (x − 3, y)

c. (x, y) → (−x, y)

4. Given the pre-image of �PQR below. Sketch the image of �PQR aft er each of the following transformations are performed.

a. refl ection over a vertical axis.

b. refl ection over a horizontal axis.

c. 90° clockwise rotation.

d. 180° rotation.

Q

P R

5. Given the line with the equation y = x - 3. Write the equation of the image of the line under the given transformation.

a. (x, y) → (x + 1, y + 3)

b. (x, y) → (x, − y)

c. 90° clockwise rotation about its y-intercept

d. 45° clockwise rotation about its y-intercept

6. MATHEMATICAL R E F L E C T I O N

How does an image compare to its pre-image

once the pre-image is refl ected over an axis of symmetry?


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes

ACTIVITY

5.2Origin-Centered DilationsAll the World’s a StageSUGGESTED LEARNING STRATEGIES: Close Reading, Activate Prior Knowledge, Create Representations, Think/Pair/Share

When performing artists travel the country to perform, they don’t bring their stages with them. Instead, they oft en contract local stagehand unions to deliver and construct platforms of various sizes and shapes at each arena. Since each event requires diff erent specifi cations, Dilation Stages off ers several confi gurations to meet the needs of the artists.

Th e stages are constructed by piecing together identical rectangular pieces to provide a given amount of surface area for the production crews to work with. To expedite this process, Dilation Stages maps out the stages on a coordinate plane to determine how many pieces of stage to load onto a truck and ship to an arena for construction.

Community Idol is holding their annual talent competition and has ordered an 8 ft × 16 ft stage to be constructed. June Yin, the shipping manager for Dilation Stages, determines what size platforms will be used to construct the stage and how many will be needed. To do this, she maps out a blueprint for the stage on a coordinate grid.

1. Plot the points A(0, 0), B(0, 2), C(4, 0), and D(4, 2) on the coordinate grid.

x

y

22

20

18

16

14

12

10

8

6

4

2

-22 4 6 8 10 12 14 16 18 20 22-2

Connect the points to form a rectangle. Th en determine the length, width, area, and perimeter of the rectangle.

ACADEMIC VOCABULARY

A dilation is a transformation that changes the size but not the shape of an object.

363-368_SB_Geom_5-2_SE.indd 363363-368_SB_Geom_5-2_SE.indd 363 1/21/10 11:55:02 AM1/21/10 11:55:02 AM

© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes

Origin-Centered Dilations ACTIVITY 5.2continued

SUGGESTED LEARNING STRATEGIES: Activate Prior Knowledge, Predict and Confirm, Create Representations, Think/Pair/Share, Quickwrite

2. Points A, B, C, and D from Item 1 will be transformed under the operation (x, y) → (2x, 2y) to generate points A′, B ′, C ′, and D′.

a. Without performing the transformation, predict the perimeter and area of the rectangle created by points A′, B ′, C ′, and D′.

b. Perform the transformation to generate points A′, B′, C′, and D′.

A′ =

B′ =

C ′ =

D′ =

c. Plot these points on the coordinate grid in Item 1, and calculate the area and perimeter of the new rectangle.

d. Compare your results to the predictions you made in part a.

All the World’s a StageAll the World’s a Stage


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


Origin-Centered Dilations

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Discussion Group, Create Representations, Look for a Pattern, Quickwrite

3. Double the x and y values of the coordinates from the second rectangle, and plot the points on the same grid as the others.

a. Complete the table below.

Width (units)

Length (units)

Perimeter (units)

Area (units2)

1st rectangle

2nd rectangle

3rd rectangle

b. Investigate the values in the table and describe any patterns you observe.

4. Since the 3rd rectangle has the exact dimensions of the Community Idol stage, June needs to determine how many 2 ft × 4 ft stage pieces need to be loaded and shipped to the arena.

a. Use a piece of patty paper to trace the outline of the 1st rectangle. Using this outline, determine how many 2 ft × 4 ft stage pieces would be needed to construct the 2nd rectangle.

b. Explain how your answer relates to the pattern of data in the chart from Item 4.



© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


SUGGESTED LEARNING STRATEGIES: Close Reading, Think/Pair/Share, Create Representations

A year has gone by, and the interest in Community Idol is waning. So, this year’s tour will be in smaller venues. Th e stage design that the producers are requesting is no longer a simple rectangle. Instead, the design is for a trapezoidal stage so that additional seating can be included as shown in the diagram below.

Seating

Seating Seating

Not wanting to lose the contract to competitors, June must fi gure out a way to design the stage. She already knows that the stage will require triangular pieces to construct, but she is unsure of what size she’ll need.

5. Th e fi nished platform will be an isosceles trapezoid with bases of 18 feet and 6 feet. Th e height of the trapezoid will be 9 feet. Use this information to plot the corners of the stage so that the bottom left vertex is located at the origin.

x

y

22

20

18

16

14

12

10

8

6

4

2

-22 4 6 8 10 12 14 16 18 20 22-2



© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


Origin-Centered Dilations

SUGGESTED LEARNING STRATEGIES: Activate Prior Knowledge, Visualization, Think/Pair/Share, Discussion Group, Quickwrite

6. June inspects the plotted stage diagram in search of a constant of proportionality that will produce an appropriate dilation.

a. Determine the smallest dilation that could be made that will produce integer values for the lengths of the bases and height of the trapezoid. Explain how you derived your answer.

b. Plot the points on the grid in Item 6, and list the coordinates of the vertices.

c. What is the constant of proportionality of this dilation?

d. Based on what you’ve learned about the relationship between areas of dilated fi gures, how many of the smaller trapezoids should fi ll the area of the actual stage? Explain your reasoning.

7. Rather than building trapezoidal platforms, the stage will be constructed from right triangular pieces. Divide the dilated trapezoid into four congruent right triangles and fi nd the total number of right triangular platforms that will be needed to construct the stage. Explain your reasoning.



© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.



SUGGESTED LEARNING STRATEGIES: Visualization, Think/Pair/Share, Discussion Group, Quickwrite

Now that Dilation Stages has begun off ering shapes other than rectangles for their stage designs, June wants to be prepared for anything customers may ask for in the future. Th e day before she leaves for vacation, an order comes in for a parallelogram stage.

8. If June starts with vertices of W(0, 0), X(1, 3), Y(5, 3), Z(4, 0) for the base platform piece, write a set of instructions that would allow her assistant to determine how many pieces would needed to build a parallelogram stage with an area of 192 square feet.



1. A right triangle has vertices A(0, 0), B(10, 0), and C(10, 24). How do the area and perimeter of the triangle change under the transformation (x, y) → (4x, 4y)?

2. Th e vertices of a dilated rectangle are P′(0, 0), Q′(18, 0), R′(18, 12), and S′(0, 12). Write the transformation notation used to fi nd P, Q, R, and S if the dilation has an area that is 9 times the original.

3. Perform the following transformations on the point (−8, 6).

a. (x, y) → (−x, −y)

b. (x, y) → (2x, 3y)

c. (x, y) → ( 3 __ 2 x, - 2 __ 3 y)

4. Triangle ABC is defi ned by vertices A(0, 0), B(15, 0), and C(15, 8).

a. Determine the coordinates of points A′, B′, and C′ under the transformation (3x, 3y).

b. Explain the diff erence in the areas of the two triangles.

5. Rectangle ROFL is defi ned by vertices R(0, 0), O(3, 0), F(3, 5), and L(0, 5).

a. Calculate the area of ROFL.

b. Write the transformation notation for ROFL → R′O′F ′L′ if R′O ′F′L′ has an area of 60 square units.

6. Th e vertices of a dilated right triangle areX′(0, 0), Y ′(10, 0), and Z ′(10, 24). Determine the transformation notation used to fi nd P, Q, and R if the dilation has an area that is 16 times the original.

7. An isosceles triangle has vertices A(0, 0), B(6, 0), and C(3, 20). How do the area and perimeter of the triangle change under the transformation (x, y) → (5x, 5y)?


How are dilations diff erent from other types of

transformations you have studied?



© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


Embedded Assessment 1 Transformations

IN MUTATIO NOS FIDES

In medieval times, a person was rewarded with a coat of arms in recognition of noble acts. A herald was commissioned to create the coat of arms.

In honor of your noble acts thus far in this course, you are being rewarded with a coat of arms. Each symbol on the grid below represents a special meaning in the history of heraldry. Th e acorn in Quadrant I stands for antiquity and strength and is also the icon used in the SpringBoard logo. Th e mascle in Quadrant II represents the persuasiveness you have exhibited in justifying your answers. Th e carpenter’s square in Quadrant III represents your conformity to the laws of right and equity (and Euclidean Geometry). Finally, the column in Quadrant IV represents the fortitude and constancy you’ve shown throughout your work in this course.

8

10

6

4

2

−8−10 −6 −4 −2 2 4 6 8 10−2

−4

−6

−8

−10

−12

−14

−16

y

x

1. To create your coat of arms, plot the outline of the shield on the coordinate grid on the grid to the right. Use the transformation (x, y) → (6x, 6y) on the points (2, 2), (−2, 2), (−2, −1), (2, −1), and (0, −3).

Use after Activity 5.2.

y

x

14

12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

-14

-16

-18

-14-12-10-8 -6 -4 -2 2 4 6 8 10 12 14

369-370_SB_Geom_U5_EA1_SE.indd 369369-370_SB_Geom_U5_EA1_SE.indd 369 1/21/10 11:22:14 AM1/21/10 11:22:14 AM

© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


Embedded Assessment 1 Use after Activity 5.2.

TransformationsIN MUTATIO NOS FIDES

2. Next, transform the fi gures from their original positions to their intended positions on the shield.

a. Refl ect the acorn over the x-axis.

b. Rotate the mascle 90° clockwise about the origin.

c. Translate the column using (x, y) → (x − 9, y + 10).

d. Rotate the carpenter’s square 90° counter-clockwise about the origin, then refl ect it over the y-axis.

Exemplary Profi cient Emerging

Representations#1, 2a-d

The student:• Plots the outline

correctly. (1)• Plots the four

fi gures correctly. (2a-d)

The student:• Plots four of

the fi ve points correctly.

• Plots three of the four fi gures correctly.

The student:• Plots only two

or three points correctly.

• Plots only one or two of the fi gures correctly.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

ACTIVITY

My Notes


5.3Tessellations Just Plane ArtisticSUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Close Reading, Graphic Organizer, Think/Pair/Share, Create Representations, Use Manipulatives, Quickwrite, Self/Peer Revision

Th e interconnection between mathematics and art is exemplifi ed by tessellations. Decorative fl oors, murals, and fabrics oft en display linked geometric designs. Beautiful geometric patterns may adorn the walls and ceilings of palaces, mosques, and temples. Th is lesson explores the mathematical properties of tessellations.

Many classrooms have fl oors and/or ceilings that are covered with square tiles. In this section, you will explore other regular polygons that could be used to completely cover fl at surfaces.

1. Demonstrate that you can cover the top of your desk with equilateral triangles in such a way that there are no gaps and no triangles overlap. Use actual equilateral triangles cut out of index cards or pattern blocks to help with this problem.

2. Describe the characteristics of an equilateral triangle that make it possible to cover the top of your desk.

You probably did not have enough triangles to actually cover the top of your desk, but you should see that given enough triangles, it would be possible. For this reason we say that equilateral triangles will tessellate by themselves.

3. Explore which other regular polygons tessellate the plane (a pure tessellation). Complete the table below.

RegularPolygon

Does ItTessellate? If Not, Describe Why Not

Triangle yesSquare yesPentagonHexagonHeptagonOctagonNonagonDecagonDodecagon

Other words used to describe tessellations include tilings and mosaics. When one or more fi gures completely cover a plane they are said to tessellate the plane.

MATH TERMS

Tessellating a plane with multiple copies of a single shape is called a pure tessellation.

MATH TERMS

ACADEMIC VOCABULARY

Covering a fl at surface with one or more types of shapes so that there are no gaps or overlaps is called a tessellation.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes

TessellationsACTIVITY 5.3continued Just Plane ArtisticJust Plane Artistic

SUGGESTED LEARNING STRATEGIES: Graphic Organizer, Think/Pair/Share, Create Representations, Look for a Pattern, Quickwrite, Self/Peer Revision

4. List each regular polygon that tessellates the plane and the measure of each interior angle.

Regular Polygon Measure of the Interior Angle

5. List each regular polygon that does not tessellate the plane and the measure of each interior angle.

Regular Polygon Measure of the Interior Angle

6. Describe what you notice about the measures of the interior angles of the polygons that tessellate the plane compared to the measures of the interior angles of the polygons that do not tessellate.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


Tessellations Just Plane ArtisticJust Plane Artistic

7. Do you think there are any other regular polygons that will tessellate by themselves? Provide an argument to support your conclusion.

In the previous items, you investigated regular tessellations. Th ey are called regular because they consist of only one type of regular polygon arranged in the same way around every point where the polygons meet. In this section, you will explore pure tessellations using non-regular polygons.

8. Consider the rectangle shown at the right. Could you create a tessellation using multiple copies of it? Explain why or why not.

9. Use an index card to draw a non-rectangular quadrilateral. Cut out your quadrilateral. On a separate sheet of unlined paper, trace multiple copies of your quadrilateral to determine if it will tessellate the plane. Draw a sketch to show the tiling or to illustrate where the quadrilaterals overlap, or have gaps. Compare your drawing to others in your class.

10. Based on what you have observed about tessellating quadrilaterals, do you believe that every triangle can be made to tessellate? Provide an argument to support your conclusion.

11. Make a conjecture concerning angle measures of non-regular polygons that tessellate. Explain your reasoning.

SUGGESTED LEARNING STRATEGIES: Predict and Confirm, Think-Pair-Share, Identify a Subtask, Look for a Pattern, Use Manipulatives, Quickwrite, Self/Peer Revision


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


SUGGESTED LEARNING STRATEGIES: Group Presentation, Think-Pair-Share, Identify a Subtask, Work Backward, Use Manipulatives, Self/Peer Revision

Th e following is an example of how you can make a fi gure that can be used to create a pure irregular translation tessellation.

A B

CD

A B

CD

A B

CD

A B

CD

Fig. 1 Fig. 2 Fig. 3 Fig. 4

12. Follow these steps to create your own unique translation tessellation.

a) Start with a rhombus on an index card with points labeled A, B, C, and D as shown above (leave margins on all sides).

b) Draw a simple curve with endpoints at A and B (Figure 1).

c) Draw a simple curve from A to D (Figure 2).

d) Cut out along the two curves and along segment DC and segment BC (Figure 2).

e) Trace the entire shape on a second index card and label points A, B, C, and D as shown (Figure 2).

f) Translate your cut fi gure down, matching points A and B to points D and C on the second index card. Trace the curve from D to C (Figure 3).

g) Translate your cut fi gure up and to the right, matching points A and D to points B and C on the second index card. Trace the curve from B to C (Figure 4).

h) Cut out the second index card along all four curves. Th is is your irregular tile to tessellate the plane (Figure 4).

i) Trace your shape to tessellate a plane (complete sheet of paper), translating it to match the edges and leaving no gaps (Figure 4).

In the fi rst part of this unit, you investigated tessellations formed by only one type of polygon. To tessellate a plane, all of the polygon angles around a vertex point have to sum to 360°. In this section, you will explore tessellations that contain more than one type of polygon.

13. Using cutouts or templates of equilateral triangles and squares, fi nd combinations that fi t around a vertex point without gaps or overlaps. Sketch your results on unlined paper.

CONNECT TO ARTART

M.C. Escher (1898–1973), from the Netherlands, was an artist who is credited with popularizing unique irregular tessellations. His artwork illustrates a deep, yet playful, understanding ofgeometric relationships.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


Tessellations Just Plane ArtisticJust Plane Artistic

SUGGESTED LEARNING STRATEGIES: Graphic Organizer, Predict and Confi rm, Group Presentation, Think-Pair-Share, Create Representations, Work Backward, Use Manipulatives, Quickwrite, Self/Peer Revision

14. Th e measure of each interior angle of an equilateral triangle is 60° and the measure of each interior angle of a square is 90°. Use this information to verify that each combination you found in Item 13 can fi t around a vertex point without gaps or overlaps.

15. Use polygon templates to discover more arrangements of regular polygons that can be arranged around a single vertex without gaps or overlaps. Sketch each arrangement on a separate sheet of paper.

16. For each arrangement you found in Item 15, use the list of interior angle measures of regular polygons, in the table below, to verify that the sum of the angles around a vertex point is 360°.

Number of Sides

Measure of Each Interior Angle

3 60°

4 90°

5 108°

6 120°

7 128 4 __ 7 °

8 135°

9 140°

10 144°

11 147 3 ___ 11 °

12 150°

In Item 15 you discovered several arrangements of regular polygons that fi t around a vertex point. Only 18 such arrangements exist. Four of them use the same polygons arranged around the vertex point in a diff erent order.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


SUGGESTED LEARNING STRATEGIES: Visualization, Group Presentation, Think-Pair-Share, Work Backwards, Use Manipulatives, Self/Peer Revision

Tessellations are named by listing the number of sides of each polygon around a vertex point, in order. For example, the polygon arrangements discovered in Item 13 could be named as follows:

Start Here

3-3-3-4-4 3-3-4-3-4

Start Here

17. Record the name of each arrangement you created in Item 15.

18. A semi-regular tessellation contains more than one type of regular polygon arranged in the same way around every vertex point. Only eight of the 18 arrangements of regular polygons repeat to form a semi-regular tessellation. Use your polygon cutouts to help you fi nd all eight of the arrangements. On a separate sheet of paper sketch enough of the tessellation to establish the pattern.

19. Using the defi nition of a semi-regular tessellation in Item 18, explain why the tessellation to the left is not a semi-regular tessellation.


Write your answers on notebook paper or grid paper.

1. Is it possible to tessellate a plane with this obtuse scalene triangle? Explain.

2. Describe the given tessellation as pureor not, regular,semi-regular, orneither, anddescribeit by its vertices.

3. What are characteristics of the polygons in the tessellation in Item 2 that guarantee there will be no overlapping or gaps?


Describe at least three diff erent tessellations you

have seen outside of a textbook or the web. State whether each tessellation is pure or not. Is it regular or semi-regular, or neither? Explain why.

See below.

Answers may vary. Sample answer: The given tessellation has two different types of vertices: 3-4-4-6 and 3-4-6-4.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

My Notes

ACTIVITY


5.4Derive and Use Area FormulasShape UpSUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Use Manipulatives, Quickwrite

Lisa works in the billing department of A Cut Above, a company that builds custom counter and table tops. A new customer has contracted A Cut Above to build the table tops for a theme restaurant: Shape Up. Each table top costs $8.50 per square foot. Lisa’s job is to calculate the area, compute the charge, and bill each customer properly.

Table tops are made with laminate delivered to A Cut Above in diff erent rectangular sizes. Before making any cuts, a cardboard template is made to use as a guide. Lisa uses the templates to investigate the areas.

One of the templates Lisa must fi nd the area of is shown at the right.

1. a. Find the area of this shape. Include units of measure in your answer. Describe the method used.

b. How much should Lisa charge the customer for this table top?

In keeping with the theme of the restaurant, the customer wants to include tables in shapes of triangles, parallelograms, trapezoids, and regular polygons. Lisa decides to investigate area formulas by fi rst fi nding the formula for the area of a parallelogram.

2. Use the rectangle provided by your teacher.

• Pick a point, not a vertex, on one side of the rectangle. Draw a segment from the point to a vertex on the opposite side of the rectangle.

• Cut along the segment.• Put the two fi gures together to form a parallelogram.

a. Explain why the area of a parallelogram is the same as the area of rectangle.

b. Nelson and Eddie are discussing how to calculate the area of any parallelogram. Nelson suggests that they multiply the lengths of two consecutive sides to fi nd the area as they do for rectangles. Eddie claims this will not work. Explain which student is correct and why.

2′

3′

2′

6′


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes

Derive and Use Area FormulasACTIVITY 5.4continued Shape UpShape Up

SUGGESTED LEARNING STRATEGIES: Marking the Text, Group Presentation, Identify a Subtask, Use Manipulatives, Quickwrite

3. A scale drawing of the parallelogram tabletop template is shown below. Determine the charge for making this table top. Show the calculations that lead to your answer.

3 ft

2 ft1.5 ft

Next Lisa decides to work with triangles. Th ere are three diff erent types of triangles that she needs to investigate.

4. Lisa decides to start her investigation with right and isosceles triangles. Sketch each of the triangles below on diff erent pieces of tracing paper. Use the traced triangles and the given triangles below to create quadrilaterals. Using what you know about these quadrilaterals, explain why the formula for the area of a triangle isArea = 1 __ 2 · base · height.

a. b.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


Derive and Use Area FormulasShape UpShape Up

SUGGESTED LEARNING STRATEGIES: Identify a Subtask, Think/Pair/Share, Quickwrite

5. Compare and contrast the steps required to fi nd the area of a right triangle and an isosceles triangle whose side lengths are given.

6. Find the area and subsequent charge for each tabletop below.

a. b. c.

2′

5′

4′ 4′

2′

6′6′

6′

7. Th e customer has also requested several tabletops in the shape of an equilateral triangle, each having diff erent side lengths. Derive an area formula that will work for all equilateral triangles with side length, s.

8. Use the formula created in Item 7 to fi nd the area of an equilateral triangle with side length 6 inches. Compare your answer to the area you found in Item 6c.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Think/Pair/Share, Create Representations, Quickwrite

9. Included in the tabletop templates is the rhombus shown at the left .

a. List the properties of a rhombus that relate to the diagonals.

b. Find the length of diagonal __

AC .

c. Apply the formula for the area of the triangles formed by the diagonals to fi nd the area of the rhombus.

10. Derive a formula for the area of a rhombus with diagonal lengths d1 and d2.

Th e area of a trapezoid with base lengths b1 and b2 and height h can be derived by applying what you have already learned about the area of a triangle.

11. Use the fi gure shown at the left to derive a formula for the area of a trapezoid. Explain how you arrived at your answer.

B C

DA

5′

5′

5′

5′ 6′

b1

b2

h


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes



SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Quickwrite

12. Bob says, “the area of a trapezoid is equal to the length of its median times its height.” Is he correct? Why?

13. Find the area of each of the trapezoids shown below.

a. 2′

5′5′ 4′

b.

9′

45°

4′

Recall that a regular polygon is equilateral and equiangular.

14. Use the tabletop template shown at the right to fi nd the area of a regular hexagon.

a. Draw all radii from the center of the circumscribed circle to each vertex of the polygon.

b. What is the measure of each of the central angles formed by the radii? Explain how you arrived at your answer.

c. Classify the triangles formed by the radii by their side length. Verify your answer.

d. How is the apothem of the hexagon related to the triangles formed by the radii?

e. Find the area of a regular hexagon with side length 4′. Show the calculations that lead to your answer.

CONNECT TO APAP

In AP Calculus, the area between curves can be approximated using trapezoids.

An apothem of a regular polygon is a perpendicular segment from a midpoint of a side of a regular polygon to the center of the circle circumscribed about the polygon.

MATH TERMS


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Create Representations, Quickwrite

15. To generalize the formula to determine the area of any regularpolygon, use the templates below.

a. How many triangles are formed when the radii are drawn from the center of the polygon to each of the vertices of the polygon with n sides?

b. What is the measure of each of the central angles formed by the radii of the n-gon? Explain how you arrived at your answer.

c. Classify the triangles formed by the radii by their side length. Verify your answer.

d. Write an expression that can be used to calculate the area of a regular n-gon with apothem length, a, and side length, s.

16. Calculate the area of a regular hexagon with apothem length 4 √

__ 3 inches and side length 8 inches.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes



SUGGESTED LEARNING STRATEGIES: RAFT (Role, Audience, Format, Topic)

17. Lisa was given the list of all of the shapes and necessary dimensions to build the remaining table tops. Remember that the charge for each top is $8.50 per square foot. Create an itemized invoice for the remaining table tops for Shape Up which includes all calculations.

Table 1: Table 2:8 ft

3 ft

6 ft7 ft

4′

4′

4′

4′

45°

Table 3: Isosceles trapezoid with base lengths 2′ and 5′ and leg lengths 2.5′

Table 4: Regular Octagon with side length 2′.

Table 5: Equilateral triangle with side lengths 3′.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.






Find the area of each shape described or shown.

1. Rectangle with base length 12 cm and diagonal length 13 cm

2. Equilateral triangle with perimeter 30 inches

3. Rhombus with diagonal lengths 6� and 10�

4. Square inscribed in a circle with radius 16 mm

5. Equilateral triangle with apothem 6 cm

6. Regular hexagon inscribed in a circle with radius 8 inches

7. 30°-60°-90° triangle with hypotenuse length 14�

8.

60° 60°

30 in.

16 in.

9. 22 cm

10 cm10 cm

45°

10.

6 mm 4 mm

11. Th e length of one diagonal of a rhombus is 10 in. and the area is 70 in2. Find the length of the other diagonal.

12. Th e area of an isosceles trapezoid is 54 square cm. Th e perimeter is 32 cm. If a leg is 7 cm long, fi nd the height of the trapezoid.


How could you use the formula for area of a

trapezoid to fi nd the area of a parallelogram?


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.



TessellationsTILE WE MEET AGAIN


Tess has been commissioned to reproduce the following design so that it can be used as a tabletop in the Escher Tessellation Museum coff ee shop.

12

3 4

1. In the sections containing the regular octagons and squares, explain the characteristics of the shapes that guarantee the design will tessellate the plane.

2. List possible angle measures for the triangles and parallelograms in the other sections that would guarantee the design will tessellate the plane. Explain.

3. To determine the production costs, Tess must know the area of each of the shapes used in the design. In regions 1 and 3, each regular octagon has side lengths of 5 inches. Find the area of each (show your calculations with answers exact or rounded to the nearest hundredth of a square inch)

• regular octagon • square • isosceles right triangle (along the edge) • isosceles right triangle (at each corner).


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.



TessellationsTILE WE MEET AGAIN

4. In regions 2 and 4, each isosceles triangle has base length 6 inches and legs 12.375 inches. Each parallelogram has side lengths 9 inches and 12.375 inches. Find the area of each (show your calculations with answers rounded to the nearest tenth of a square inch)

• isosceles triangle • parallelogram • right triangle (along the edge)


Math Knowledge#1, 2, 3, 4

The student:• Describes

the correct characteristics of the two shapes. (1)

• Lists correct angle measures that make it possible for triangles and parallelograms to tessellate. (2)

• Finds the correct areas, rounded correctly, of each of the four fi gures. (3)

• Finds the correct areas, rounded correctly, of each of the three fi gures. (4)

The student:• Describes only

one characteristic of a shape that will tessellate.

• Lists correct angle measures for either triangles or parallelograms that make it possible for them to tessellate.

• Uses the correct method to fi nd three or four of the areas but makes a computational error.

• Uses the correct method to fi nd two or three of the areas but makes a computational error.

The student:• Is not able

to describe a characteristic.

• Lists the correct angle measures for neither triangles nor parallelograms that make it possible for them to tessellate.

• Finds one or two correct areas.

• Finds only one correct area.

Communication#1, 2

The student:• Writes a

complete explanation of the characteristics that guarantee the design will tessellate. (1)

• Writes a complete explanation about the angle measures that would make it possible for both the triangles and parallelograms to tessellate. (2)

The student:• Writes an

incomplete explanation of the characteristics.

• Writes a complete explanation about the angle measures for either the triangles or the parallelograms.

The student:• Writes an

incorrect explanation of the characteristics.

• Writes an incorrect explanation about the angle measures for both fi gures.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes

ACTIVITY

5.5Non-Euclidean GeometryIsn’t That Spatial?SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Questioning the Text, Shared Reading, Visualization, Think/Pair/Share, Create Representations, Look for a Pattern, Use Manipulatives, Quickwrite, Self/Peer Revision

IntroductionTh is geometry course has explored the relationships among geometric fi gures such as lines, angles, polygons, circles, and three-dimensional objects. An axiomatic system has been used to develop a logical sequence of theorems. As those of you who have been to the moon (and all who have seen the photos) realize, the world is not fl at! Yet much of what we studied assumed a fl at-plane world. Euclidean properties suffi ce for geometric applications faced by the average person. But for space agencies to perform calculations necessary to get the rover to land on Mars in May 2008, spherical geometry was a must! Mathematicians have pondered the existence of non-Euclidean geometries since the early 1800s. Research in these areas continues to be a considerable focus of study today.

Spherical geometry is one branch in the non-Euclidean fi eld. It explores the geometric characteristics of fi gures on the surface of a sphere. Th ere are signifi cant diff erences in properties with this change in perspective. It is helpful to have a globe or ball to contemplate these ideas.

1. Choose a point on the sphere provided by your teacher. Imagine that an ant walks on the sphere in a straight line so that it does not fall off .

a. Trace the path that the ant would follow.

b. Describe the path walked by the ant.

c. Use a rubber band or masking tape to mark the path of the ant.

2. Choose a diff erent point on the sphere. Imagine that a second ant walks in a straight line trying to avoid the path of the fi rst ant. Mark the path of the second ant.

3. How can the second ant choose a path that will not cross the path of the fi rst ant?

Great circles are the largest possible circles drawn on a sphere.

MATH TERMS


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes

Non-Euclidean Geometry ACTIVITY 5.5continued Isn’t That Spatial?Isn’t That Spatial?

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Close Reading, Questioning the Text, Visualization, Create Representations, Think/Pair/Share, Look for a Pattern, Use Manipulatives, Quickwrite, Self/Peer Revision

4. In Euclidean geometry, if two lines intersect, they intersect in exactly one point. In spherical geometry, what is true about any two lines?

5. In spherical geometry, a line is defi ned to be a great circle of the sphere. Lines of longitude on a globe are examples of these. Why do you suppose that airlines design fl ight paths along spherical lines?

Antipodal points (also called antipodes) are two points that are diametrically opposite. A straight line through the center of the Earth connects them.

6. Locate the antipode for your city. State its location.

7. Explain why most points on Earth have oceanic antipodes.

8. Choose a point on your sphere.

a. Draw two great circles through the point.

b. Use a protractor to measure the angles formed at the given point on the sphere by the great circles.

c. What do you notice about the measures of the angles formed by the great circles?

A lune is a region bounded by two great circles.

The word is derived from the Latin word for moon.

MATH TERMS


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes

SUGGESTED LEARNING STRATEGIES: Questioning the Text, Visualization, Group Presentation, Think/Pair/Share, Create Representations, Look for a Pattern, Quickwrite, Self/Peer Revision


Non-Euclidean Geometry Isn’t That Spatial?Isn’t That Spatial?

9. On the sphere provided by your teacher, choose a point N on your sphere.

a. Draw a great circle through point N.

b. Draw a second great circle through point N to create a lune with a 90° angle.

c. Draw a great circle that does not contain point N to divide the lune into two congruent regions.

d. How many non-overlapping sections are created by theintersections of the three great circles?

e. Each section is a spherical triangle. Use a protractor to measure the three angles of a triangle.

f. What is the sum of the measures of those three angles?

10. Choose any three random non-collinear points on the sphere. Draw the triangle through the points. Remember to draw the segments along a great circle.

11. Measure the three angles of the triangle and determine the sum of their measures.

12. Use the given diagram with triangle ABC.

a. What happens to the sum of the measures of the angles in the triangle as points B and C come closer and closer together?

b. Based on your answer in part a, what is the least possible sum for the three angles of a spherical triangle?

AC

B


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes

Non-Euclidean Geometry ACTIVITY 5.5continued Isn’t That Spatial?Isn’t That Spatial?

SUGGESTED LEARNING STRATEGIES: Visualization, Think/Pair/Share, Look for a Pattern, Quickwrite, Self/Peer Revision

13. Use the given diagram with triangle DQT.

a. What happens to the measure of each angle in the triangle as the points move outward and approach the same great circle?

b. Based on your answer in part a, what is the greatest possible sum for the three angles of a spherical triangle?

14. Write the possible measurements for the sum of the measures of the angles of a triangle as a compound inequality.


Write your answers on notebook paper.

1. Are the lines of the Earth’s latitude considered lines in spherical geometry? Explain.

2. Locate two land-based antipodes on Earth.


Compare and contrast Euclidean geometry and

spherical geometry.

4. Why do you suppose Euclidean geometry is still considered to be an essential component of the high school curriculum?

DQ

T

As the points move outward and approach the same great circle, the measures of each of the three angles approach 180°.

The greatest possible sum for the three angles of a spherical triangle is a measure slightly less than 540°.

180° < x < 540°


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes

ACTIVITY

5.6VectorsPoint the WaySUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Marking the Text, Discussion Group, Think/Pair/Share, Create Representations, Quickwrite

A river is fl owing in the direction south to north. Amanda and Brody want to cross the river, in their canoe, from west to east. Th e illustration to the right gives a top view of the river and the canoe. Th e arrows indicate the direction of movement for the water and the canoe.

1. Describe the eff ect that the fl owing water will have on the movement of the canoe.

Th e movement of the water and the boat can be represented by vectors. A vector is a quantity that has both direction and magnitude (length). In this case, the magnitude of the vector represents the speed of the water or boat and the direction of the vector indicates the direction of the water or boat.

2. Th e water in the river is fl owing from south to north at 4 miles per hour. On the graph below, sketch a vector to represent the movement of water in the river.

54321

-1-2-3-4-5

-5-4-3-2-1 1 2 3 4 5

Vectors with the same direction and magnitude are said to be equivalent.

3. On the coordinate grid above, graph and give the terminal point of the vectors equivalent to your original vector whose initial point is given below.

a. (1, -2) b. (-4, 1)

4. Amanda claims that the vector with initial point (-3, 3) andterminal point (-3, -1) is equivalent to the vectors drawn on your graph. Contradict or defend her claim.

RIVER

Canoe

ACADEMIC VOCABULARY

vector

Vectors are represented with directed line segments from an initial point to a terminal point. The direction is indicated with an arrow at the terminal point. If P is the initial point and Q is the terminal point, then the vector is written as � � PQ . The magnitude of � � PQ is the same as the length of

__ PQ .

MATH TERMS


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


t

My Notes

SUGGESTED LEARNING STRATEGIES: Visualization, Marking the Text, Think/Pair/Share, Create Representations, Quickwrite, Self/Peer Revision

Vectors ACTIVITY 5.6continued Point the WayPoint the Way

5. Translate the vector from Item 3a so that the initial point is (0, 0). Give the coordinates of the new terminal point and describe the translation you used.

6. Translate the vector from Item 3b so that the initial point is (0, 0). Give the coordinates of the new terminal point and describe the translation you used.

7. Compare and contrast your responses in Items 5 and 6.

A vector whose initial point is (0, 0) is in standard position. In standard position the vector can be represented by the coordinates of the terminal point. Th is is called component form ⟨v1, v2 ⟩.

8. Write the component form of the vector that represents the water in the river.

9. Amanda and Brody can row the canoe 3 miles per hour in still water.

a. On the grid in the My Notes section, sketch a vector in standard position to represent the movement of the canoe.

b. Write the component form of the vector that represents the move-ment of the canoe.

TRY THESE A

Let � � PQ be a vector with P(1, 2) and Q(4, 7).

a. On the grid at the top of the My Notes section of the next page, graph � � PQ .

WRITING MATH

Vectors in standard position are written with one lower case letter � � v .

CONNECT TO APAP

Vectors are used in AP Calculus to represent the position, velocity, and acceleration of an object moving in the two-dimensional coordinate plane.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations, Quickwrite

My Notes


VectorsPoint the WayPoint the Way

b. Calculate the terminal point of the equivalent vector in standard position.

c. Write the component form of � � PQ .

d. Determine the component form of the vector with initial point (-2, 3) and terminal point (4, -1).

Th e actual speed of the canoe can be represented by the sum of the water and canoe vectors, � � w and � � c .

10. Follow the steps below to geometrically add the two vectors. Use the coordinate grid at the right.

a. Graph vectors, � � w and � � c , in standard position.

b. Draw the vector equivalent to � � w so that the initial point is the terminal point of � � c .

c. Draw the vector equivalent to � � c so that the initial point is the terminal point of � � w .

d. Th e sum, � � w + � � c , is the diagonal of the parallelogram with initial point at (0, 0). Draw the vector that represents the sum.

11. What is the component form of the vector that represents the actual speed of the canoe in the water?

12. Explain how to determine the sum of the vectors algebraically.

13. Using the vectors � � v1 = ⟨a, b⟩ and � � v2 = ⟨c, d ⟩, write an expression for � � v1 + � � v2 .

WRITING MATH

The magnitude of a vector, � � v , can be denoted using | � � v |.

87654321

-1-2

-5-4-3-2-1 1 2 3 4 5

654321

-1-1 2 3 4 5 61


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite

t

My Notes

Vectors ACTIVITY 5.6continued Point the WayPoint the Way

14. Sketch the vector, � � w + � � c , from Item 10 on the coordinate grid to the left .

15. Use the Pythagorean Th eorem to calculate the magnitude of the vector, � � w + � � c . Explain the meaning of the magnitude in terms of the movement of the canoe. Include units of measure with your answer.

16. Using the vector � � v = ⟨a, b⟩ write an expression that can be used to calculate the magnitude of the vector.

TRY THESE B

a) Given � � u = ⟨-1, 5⟩. Calculate | � � u |.

b) Given � � AB with A(2, -4) and B(-1, 0). Find | � � AB |.

CHECK YOUR UNDERSTANDINGWrite your answers on notebook paper or on grid paper. Show your work.

1. A vector � � v has initial point (-2, 7) and terminal point (3, 4).

a) Graph a vector equivalent to � � v that has initial point (1, 5). What are the coordinates of the terminal point?

b) Determine the component form of � � v .

c) Find | � � v |.

2. An airplane is fl ying northeast. Its direction and speed can be represented by the vector � � p = ⟨500, 500⟩. Th e airplane encounters wind blowing 100 miles per hour from east to west.

a) Find the speed of the airplane before it meets the wind.

b) Write the component form of the vector representing the wind, � � w .

c) Calculate � � p + � � w and use it to find the speed of the plane in the wind.


Why would it be important for a jet

pilot to understand vectors?

654321

-1-1 2 3 4 5 61


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

My Notes

ACTIVITY


5.7Transformations with MatricesMoved by MatricesSUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Marking the Text, Summarize/Paraphrase/Retell, Think/Pair/Share, Create Representations

Instead of using real events, movie and television special eff ects are oft en computer generated. Oft en these eff ects are created through the use of simple images enhanced with transformations or complex images broken down into polygons which are then transformed. Computers use matrices to translate, refl ect, and dilate images.

A matrix is a rectangular array of numbers. Matrices are described using the number of rows fi rst and then the number of columns. Th ese are the

dimensions of the matrix. For example, ⎡ ⎢

⎣ 2 0

−4 1

3 −5 6

7 ⎤ �

⎦ is a 2 × 4 matrix.

Ordered pairs (x, y) can be expressed using a column matrix in the

form ⎡

⎢

⎣ x y

⎤ �

⎦ .

1. Write a column matrix for the ordered pair (−3, 5) and state the dimensions.

In a polygon, the ordered pairs that represent the vertices can all be placed into the columns of a matrix, resulting in a vertex matrix.

2. Th e quadrilateral ABCD has vertices A(−3, −2), B(0, 2), C(3, 4) and D(0, −1).

a. Write a vertex matrix for ABCD.

b. Label each column with the vertex it represents.

c. What does each row in your vertex matrix represent?

3. State the type of polygon and the vertices represented in the following vertex matrix.

⎡

⎢

⎣ 0 4

1 5 4 1

6 3

4 6

⎤ �

⎦

A column matrix is a matrix with only one column.

MATH TERMS

A vertex matrix is a matrix containing the vertices of a polygon.

MATH TERMS

ACADEMIC VOCABULARY

matrix


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

My Notes


Transformations with Matrices ACTIVITY 5.7continued Moved by MatricesMoved by Matrices

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations

Transformations of polygons can also be represented by matrices.

4. Use pentagon ABCDE below to answer parts a–h.

a. Write the vertex matrix for pentagon ABCDE (preimage).

b. Give the matrix that results from adding 3 to each x-coordinate and −4 to each y-coordinate.

c. State the vertices represented in this new matrix.

d. Graph and label the resulting polygon (image) on the same set of axes as pentagon ABCDE.

e. What type of transformation took place to create the image?

f. What dimensions should any matrix added to your vertex matrix have?

g. Write the matrix that was added to the vertex matrix of pentagon ABCDE to create the image.

h. Write the matrix that would be added to the vertex matrix of pentagon ABCDE to create an image 5 units to the left and 3 units up from the preimage.

To be added or subtracted, two matrices must have the same dimensions.

⎡ ⎢

⎣

2

−1

4

0

−3

5

⎤ �

⎦ +

⎡ ⎢

⎣ 1

4

3

−5

2

0 ⎤ �

⎦

= ⎡

⎢

⎣ 3

3

7

−5

−1

5

⎤ �

⎦

A

B

C

D

E2 4 6 8 10

8

6

4

2

-2

-4

-6

-8


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

My Notes



Transformations with Matrices Moved by MatricesMoved by Matrices

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations

5. Use �GHI shown to the right to answer parts a–g.

a. Write the vertex matrix for �GHI.

b. Multiply each element of the vertex matrix by 2.

Write the new matrix.

c. State the vertices represented in this new matrix.

d. Graph the image on the same set of axes as preimage �GHI.

e. What type of transformation took place to create the image?

f. What is the relationship between the perimeter of the image and the perimeter of the preimage?

g. What would need to be done to the vertex matrix of preimage �GHI to create an image with a perimeter one–half as large as that of �GHI?

An element of a matrix is a number in the matrix.

MATH TERMS

G I

H

10

8

6

4

2

−22 4 6 8 10


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

My Notes



SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations, Look for a Pattern

6. Use �MNO to answer parts a–g.

8

10

M

N

O

6

4

2

−8−10 −6 −4 −2 2 4 6 8 10−2

−4

−6

−8

−10

y

x

a. Write the vertex matrix for �MNO

b. Multiply the matrix ⎡ ⎢

⎣ 1 0

0 −1

⎤ �

⎦ by the vertex matrix. Write the

resulting matrix.

c. Which numbers in the product matrix stayed the same?

d. Which numbers in the product matrix changed?

e. Graph the image on the same set of axes as preimage �MNO.

f. What type of transformation took place to create the image? Be specifi c.

A graphing calculator can be used to multiply matrices.

TECHNOLOGY


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes



SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations, Look for a Pattern, Quickwrite, Self/Peer Revision

7. Use �MNO to answer parts a–e.

8

10

M

N

O

6

4

2

−8−10 −6 −4 −2 2 4 6 8 10−2

−4

−6

−8

−10

y

x

a. Multiply the matrix ⎡ ⎢

⎣ −1 0

0 1

⎤ �

⎦ by the vertex matrix. Write the

resulting matrix.

b. Which numbers in the product matrix stayed the same?

c. Which numbers in the product matrix changed?

d. Graph the image on the same set of axes as preimage �MNO.

e. What type of transformation took place to create the image? Be specifi c.

8. Compare and contrast the matrices used to transform the vertex matrices and the product matrices in items 6b and 7a. Explain how the placement of the −1 aff ected the transformation of each of the preimages.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Create Representations, Look for a Pattern, Identify a Subtask

9. Use �PQR to answer parts a–d.

8

10

P R

Q

6

4

2

−8−10 −6 −4 −2 2 4 6 8 10−2

−4

−6

−8

−10

y

x

a. Write the vertex matrix for �PQR.

b. Use matrix multiplication to refl ect �PQR in the x-axis. State the matrix you used for this refl ection and graph the image. Label the image �P'Q'R'.

c. Use matrix multiplication to refl ect �P'Q'R' in the y-axis. State the matrix you used for this refl ection and graph the image. Label the image �P"Q''R''.

d. Th e transformation from �PQR to �P''Q''R'' can be accomplished in one step by multiplying the preimage by a single matrix. Write the matrix you think should be used. Justify your answer.

The combination of two refl ections, one over the x-axis and one over the y-axis is also referred to as a refl ection over the origin.

MATH TERMS

WRITING MATH

Recall that prime and double prime notation indicate cor-responding parts of congruent or similar fi gures. In �PQR and �P'Q'R', ∠P' corresponds to ∠P, ∠Q' corresponds to ∠Q and ∠R' corresponds to ∠R. In �P'Q'R' and �P"Q"R", ∠P" corresponds to ∠P', ∠Q" corresponds to ∠Q' and ∠R" corresponds to ∠R'.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes



10. Multiply each of the following matrices by the vertex matrix of �EFG, graph the image, and describe the direction and angle of rotation about the origin that results from each.

a. ⎡ ⎢

⎣ 0 1

-1 0

⎤ �

⎦

b. ⎡ ⎢

⎣ -1 0

0

-1 ⎤ �

⎦

c. ⎡ ⎢

⎣

0 -1

1 0

⎤ �

⎦

11. Use your answers from Items 4–10 to determine what matrix operation is needed to perform each of the following transformations on a 2 × 4 vertex matrix for a quadrilateral. Be specifi c.

a. dilation by a scale factor of 5 b. refl ection in the origin

c. rotation 90° clockwise about the origin

d. refl ection over the x-axis

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share

8

10

G

F E

6

4

2

−8−10 −6 −4 −2 2 4 6 8 10−2

−4

−6

−8

−10

y

x


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

My Notes



e. rotation 180° counterclockwise about the origin

f. translation right 5 units and down 2 units

g. rotation 270° clockwise about the origin

h. refl ection over the y-axis

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share



1. A game designer is creating a graphic by transforming various polygons. He begins with �GHI having G(-3, 5), H(4, 8), and I(7, 5).

a. Use a matrix to fi nd the coordinates of the image aft er he rotates the triangle 90° clockwise about the origin. Show the matrices you used.

b. What would be the coordinates of the image if he decided to enlarge the preimage to 3 times its initial size and then translate it up 2 units and 5 units left ? Show the matrices that you used to fi nd your answer.

2.

a. Write a matrix that could be used to

refl ect a polygon over the line y = x. Explain your answer.

b. Write a matrix that could be used to refl ect a polygon over the line y = –x. Explain your answer.

MATHEMATICAL R E F L E C T I O NMATHEMATICAL R E F L E C T I O N


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.



Matrices, Transformations, and VectorsFISHING FOR MATRICES


1. Rectangle ABCD has vertices A(3, 0), B(3, 2), C(7, 2) and D(7, 0).

a. Write the vertex matrix for rectangle ABCD.

b. Write the matrix for a dilation of 5.

2. Write the matrix that will translate �ABC ⎡

⎢

⎣ 1 0

2 -3

4 -1

⎤ �

⎦ so that B' is located at

the point (1, -1). Justify your response.

3. A river is fl owing in the direction west to east at 5 miles per hour. R. Flowing wants to cross the river in his fi shing boat from south to north. His boat can travel 12 mph in still water.

a. Sketch vectors, in standard position, to represent the water, w , and the boat in still water, b . Label each vector.

b. Determine the component form for each vector in part a.

c. Determine the vector w + b .

d. Find the actual speed of the boat in the moving water. Be sure to include units with your answers.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.



Matrices, Transformations, and VectorsFISHING FOR MATRICES


Math Knowledge#1a, b; 3b, c

The student:• Writes the

correct matrices. (1a, b)

• Determines the correct component form for both vectors. (3b)

• Determines the correct sum of the vectors. (3c)

The student:• Writes only one

correct matrix.• Determines

the correct component form for only one vector.

• Uses the correct method to fi nd the sum but makes a computational error.

The student:• Writes both

matrices incorrectly.

• Determines neither correct component form.

• Determines an incorrect sum.

Problem Solving#2, 3d

The student:• Writes the

correct matrix that will translate the triangle’s coordinates. (2)

• Finds the correct actual speed of the boat. (3d)

The student:• Writes a matrix

with some correct entries.

The student:• Writes a matrix

with no correct entries.

• Finds the incorrect speed.

Representations#3a, d

The student:• Sketches the

correct vectors. (3a)

• Includes the correct units with the answer. (3d)

The student:• Sketches only

one of the vectors correctly.

The student:• Sketches neither

of the vectors correctly.

• Omits the units with the answer.

Communication#2

The student writes a complete justifi cation for the translated matrix. (2)

The student writes an incomplete justifi cation for the matrix.

The student gives an incorrect justifi cation for the matrix.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

UNIT 5Practice


ACTIVITY 5.1

1. Which transformation would reflect a pre-image over the x-axis and shift it to the right 5 units?

a. (x, y) → (x, 5 - y) b. (x, y) → (5 - x, y) c. (x, y) → (-x, y + 5) d. (x, y) → (x + 5, -y)

2. Given the line with equation y = -x. Write the equation for the image of the line under the given transformation.

a. (x, y) → (x, y - 2) b. (x, y) → (4 - x, y) c. 90° clockwise rotation about the origin d. 45° clockwise rotation about the origin

3. Let �ABC be the pre-image with coordinates A(0, 0), B(6, 0), and C(3, 3 √

__

3 ). Find the coordinates of the vertices of the image, �A'B'C', once the following transformations have been performed.

a. (x, y) → (x - 3, y + 2) b. (x, y) → (x, 2 - y) c. �ABC is rotated 60° clockwise about point A d. �ABC is rotated 30° counterclockwise about

point A.

ACTIVITY 5.2

4. Perform the following transformations on the point (3, -5).

a. (x, y) → (4x, -y) b. (x, y) → (-5x, y) c. (x, y) → ( 1 __ 2 x, -

4 __ 5 y)

5. Trapezoid QRST has vertices Q(0, 0), R(16, 0), S(12, 8), and T(4, 8). Find the area of a dilated image of the trapezoid under a transformation

scale factor of 1 __ 3 .

6. An isosceles triangle has vertices A(0, 0), B(9, 0), and C(4.5, 16). How do the area and perimeter of the triangle change under the transformation (x, y) → 2 __ 5 (x, y)?

7. Rectangle TUVW has vertices T(0, 0), U(4, 0), V(4, 3), and W(0, 3). Rectangle T'U'V'W' has been dilated so that it has an area that is 36 times the area of TUVW. Determine the vertices of T'U'V'W'.

ACTIVITY 5.3

8. Is it possible to tessellate a plane with the given scalene quadrilateral? Explain.

9. Describe the given tessellation as pure or not, regular, semi-regular, or neither, and describe it by its vertices.

10. What are characteristics of the polygons in the tessellation in Item 9 that guarantee there will be no overlapping or gaps?

405-407_SB_Geom_U5_PRAC_SE.indd 405405-407_SB_Geom_U5_PRAC_SE.indd 405 1/21/10 11:26:27 AM1/21/10 11:26:27 AM

© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

UNIT 5 Practice


ACTIVITY 5.4

Find the area of each polygon described below.

11. Isosceles triangle with side lengths 15 ft , 15 ft , 18 ft

12. Regular pentagon inscribed in a circle with radius 8 ft

13. Rectangle that is 9 cm long and has a perimeter that is 26 cm

14. Equilateral triangle with apothem 10 in.

15. Isosceles trapezoid with legs 10 cm and base lengths 12 cm and 28 cm

16. 45°-45°-90° triangle with hypotenuse length 10 cm

17.

120°

120° 60°

60°

16 mm

18 mm

18.

4 mm 8 mm

19. Th e length of one diagonal of a rhombus is 7 m and the area is 42 m2. Find the length of the other diagonal.

20. Th e area of a trapezoid is 120 cm2. Find the height if the median is 24 cm. long.

ACTIVITY 5.6

21. A vector � � v has initial point (2, -8) and terminal point (5, -7).

a. Determine the component form of � � v b. Find | � � v |.

22. Determine which vector has magnitude 1.

a. ⟨ -5 ____ √

___

21 , 2 ____

√

___

21 ⟩ b. ⟨ -1, 1⟩

c. ⟨ √

__ 5 , 1 ___

√

__ 5 ⟩ d. ⟨

5 ____ √

___

29 , -2 ____

√

___

29 ⟩

23. Given � � u = ⟨ 2, -3⟩ and � � v = ⟨ 1, 5⟩ a. show � � u + � � v graphically. b. find � � u + � � v numerically.

24. An airplane is fl ying due north (from south to north) at a constant speed of 680 mph. Th e plane encounters a headwind. Th e speed of the wind can be represented by the vector, � � w , ⟨ 14, -78⟩ .

a. Write the component form of the vector representing the airplane, � � p .

b. Find the speed of the wind. c. Calculate � � p + � � w and use it to find the speed

of the plane in the wind.

ACTIVITY 5.7

25. Which translation matrix could be used to translate �DEF six units to the right and nine units up?

a. ⎡ ⎢

⎣ 6 9

6 9

6 9

⎤

⎦

b. ⎡

⎢

⎣ -6 9

-6 9

-6 9

⎤

⎦

c. ⎡

⎢

⎣ 6

-9

6 -9

6

-9 ⎤

⎦

d. ⎡

⎢

⎣ -6 -9

-6 -9

-6 -9

⎤

⎦


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

UNIT 5Practice


26. �XYZ is represented by ⎡ ⎢

⎣ -2 -4

1

-6

3 5 ⎤

⎦ . Find

the vertex matrix for the image aft er a clockwise 180° rotation. Show your work.

27. Use matrices to find the coordinates of the image of parallelogram WXYZ with W(-8, 1), X(-2, 3), Y(-1, 0), and Z(-6, -3) after a rotation 90° counterclockwise followed by a dilation with a scale factor of -

1 __ 2 . Show your work.

28. Use matrices to fi nd the coordinates of the image of �ABC with A(2, 3), B(4, -1), and C(-2, -3) aft er a dilation with a scale factor of 2 followed by a refl ection across the x-axis.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

UNIT 5 Reflection


An important aspect of growing as a learner is to take the time to refl ect on your learning. It is important to think about where you started, what you have accomplished, what helped you learn, and how you will apply your new knowledge in the future. Use notebook paper to record your thinking on the following topics and to identify evidence of your learning.

Essential Questions

1. Review the mathematical concepts and your work in this unit before you write thoughtful responses to the questions below. Support your responses with specifi c examples from concepts and activities in the unit.

What connections exist between transformations and dilations and congruence and similarity?

How are transformations and tessellations used in real world settings?

Academic Vocabulary

2. Look at the following academic vocabulary words:rotation dilation matrixvector tessellation

Choose three words and explain your understanding of each word and why each is important in your study of math.

Self-Evaluation

3. Look through the activities and Embedded Assessments in this unit. Use a table similar to the one below to list three major concepts in this unit and to rate your understanding of each.

Unit Concepts

Is Your Understanding Strong (S) or Weak (W)?

Concept 1

Concept 2

Concept 3

a. What will you do to address each weakness?

b. What strategies or class activities were particularly helpful in learning the concepts you identifi ed as strengths? Give examples to explain.

4. How do the concepts you learned in this unit relate to other math concepts and to the use of mathematics in the real world?

408_SB_Geom_5_SR_SE.indd 408408_SB_Geom_5_SR_SE.indd 408 1/21/10 11:26:46 AM1/21/10 11:26:46 AM

© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


Unit 5

Math Standards Review

1. When tiling the fl oor in her bathroom, Maria decides to use just one type of regular polygon tile. Which shape should she use so that there are no empty gaps between the tiles?

A. pentagon

B. hexagon

C. octagon

D. decagon

2. Romi’s above-ground pool is in the shape of a regular octagon with side lengths of 6 feet. She wants to put a cover over the pool when not in use. How much material, to the nearest square yard, should Romi buy to cover the top of her pool?

3. Madigal’s rectangular bedroom measures 10 feet by 11 feet. His parents’ rectangular bedroom measures 20 feet by 22 feet. What is the ratio of the area of his bedroom to the area of his parents’ bedroom?

1. Ⓐ Ⓑ Ⓒ Ⓓ

2.

○‒ ⊘⊘⊘⊘○• ○• ○• ○• ○• ○•⓪⓪⓪⓪⓪⓪①①①①①①②②②②②②③③③③③③④④④④④④⑤⑤⑤⑤⑤⑤⑥⑥⑥⑥⑥⑥⑦⑦⑦⑦⑦⑦⑧⑧⑧⑧⑧⑧⑨⑨⑨⑨⑨⑨

3.

○‒ ⊘⊘⊘⊘○• ○• ○• ○• ○• ○•⓪⓪⓪⓪⓪⓪①①①①①①②②②②②②③③③③③③④④④④④④⑤⑤⑤⑤⑤⑤⑥⑥⑥⑥⑥⑥⑦⑦⑦⑦⑦⑦⑧⑧⑧⑧⑧⑧⑨⑨⑨⑨⑨⑨

409-410_SB_Geom_U5_MSR_SE.indd 409409-410_SB_Geom_U5_MSR_SE.indd 409 1/21/10 11:59:07 AM1/21/10 11:59:07 AM

© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


Unit 5 (continued)

Math Standards Review

4. An airplane takes off from Orlando International Airport heading due north at 500 miles an hour. Th e plane encounters constant wind moving from west to east at 30 miles an hour.

Part A: Write each vector in the form ⟨v1, v2⟩.

• Write a vector that represents the airplane.

• Write a vector that represents the wind.

• Write a vector that represents the direction in which the airplane is actually fl ying relative to the ground.

Part B: Aft er one hour, how far has the airplane traveled, to the nearest mile? Show your work or give a written explanation to justify your answer.

Read

Explain

Solve

409-410_SB_Geom_U5_MSR_SE.indd 410409-410_SB_Geom_U5_MSR_SE.indd 410 1/21/10 11:59:08 AM1/21/10 11:59:08 AM

Documents

Coordinate 5 Geometry and Transformationsclassroom.dickinsonisd.org/users/0315/docs/cb_sbo_math_miu_l5_u5... · Unit5 349 Coordinate ... Embedded Assessment 1 Transformations p. 369