similarity activity similari-teen saves the day 3-+unit+3... · similari-teen saves the day ... e...

© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

Unit 3 • Similarity, Right Triangles, and Trigonometry 205

My Notes

ACTIVITY

3.2SimilaritySimilari-Teen Saves the DaySUGGESTED LEARNING STRATEGIES: Close Reading, Marking the Text, Questioning the Text, Role Play, Shared Reading, Summarize/Paraphrase/Retell, Think Aloud, Visualization

For a Geometry class project, Tristan and Shawnda are creating a game based on properties of similarity. ! eir game is a tabletop role playing game (RPG). A typical RPG involves players who create " ctional characters to participate in imaginative stories. ! ese characters typically have speci" c abilities, which they use throughout the story to drive the action and outcome of the game. ! e use of these abilities is de" ned by a formal system of rules which is governed by a designated game master, or GM, for the session.

! e game that Tristan and Shawnda created involves superheroes in an ongoing " ght for justice against the sinister Dr. Protractor. In each story their characters face challenges that can be completed by applying the properties of similarity. However, within the rules of the game, they must " rst prove the theorems involved in each scenario before applying their abilities to complete the tasks.

! e AA Similarity Postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

1. Use the triangle angle sum theorem to explain why it is not necessary to show that all three pairs of corresponding angles are congruent.

During their classroom presentation of the game, Shawnda, the GM for the story, presents Tristan with the following scenario.

GMYou’ve reached Dr. Protractor’s lab. As you peer through the keyhole, a curtain is blocking your view. On the wall to the right, a mirror reveals the re! ection of a strange device in the back le" corner of the room. You must destroy this device. What will you do next?

Similari-TeenI will open the door.

GMDr. Protractor may be foolish enough to think that 1 is a prime number, but he knows to keep his door locked. Try again, Similari-Teen.

Similari-TeenI will re! ect the rays of my heat vision o# the mirror and destroy the device.

GMVery well, Similari-Teen. Your $ rst task is to explain the Angle-Angle (AA) Similarity Postulate, prove the Side-Angle-Side (SAS) Similarity % eorem, and apply the Side-Side-Side (SSS) Similarity % eorem.

205-216_SB_Geom_3-2_SE.indd 205205-216_SB_Geom_3-2_SE.indd 205 1/21/10 2:16:16 PM1/21/10 2:16:16 PM

© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.

206 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Similarity ACTIVITY 3.2continued Similari-Teen Saves the DaySimilari-Teen Saves the Day

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Debriefing, Think/Pair/Share, Quickwrite

2. Consider !DEF and !GHF shown in the My Notes section.

a. Which pair of angles are marked congruent in the diagram?

b. What other pair of angles are congruent? Explain your reasoning.

c. Similarity Statement: !DEF ~ ! by the AA Similarity Postulate.

GMWell done, Similari-Teen. Now you must prove the SAS Similarity ! eorem.

! e Side-Angle-Side (SAS) Similarity ! eorem states that if an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar.

3. In triangles QRS and VTU, ∠Q # ∠V and QR ___ VT = QS ___ VU .

a. Con" rm that the sides including ∠Q and ∠V are in proportion.

b. Draw point W on __

TV so that __

WV # __

RQ .

c. ! rough point W, draw a line parallel to __

TU . Label the intersection of the line and

__ UV as point X.

d. Explain how you can now prove that !TUV ∼ !WXV.

e. What can you conclude about !RSQ and !WXV ? Explain your reasoning.

E

D

G

H

F

T

V

U

72 81

R

Q

S

32 36


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes

ACTIVITY 3.2continued

Similarity Similari-Teen Saves the DaySimilari-Teen Saves the Day

SUGGESTED LEARNING STRATEGIES: Close Reading, Debriefing, Create Representations, Discussion Group, Think/Pair/Share, Quickwrite

4. Use the information you gathered in Item 3 to construct a formal proof of the SAS Similarity ! eorem.

GMExcellent work, Similari-Teen. Now you must apply the Side-Side-Side Similarity ! eorem in order to proceed with foiling Dr. Protractor’s plan.

! e Side-Side-Side (SSS) Similarity ! eorem states that if the corresponding sides of two triangles are in proportion, then the triangles are similar.

5. Help Similari-Teen complete his " nal task by solving for x in the following problem.Given: !QRS ˜ !TVU

R

TU

V

8 cm x + 4 cm

Q

S

14 cm3x – 5 cm

GMSince this will be the " rst time you’ve used your heat vision, your next task will be to model the situation before attempting your hit. You’ve only got one shot at this, Similari-Teen, so you must get it right the " rst time. I suggest enlisting several allies before proceeding any further.

The proof of SSS Similarity Theorem can be done using a method similar to the proof of the SAS Similarity Theorem and is left as Exercise 4 in Check Your Understanding at the end of this activity.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


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

6. Locate a spot on the ! oor that is 20 feet away from one of the walls of the classroom. Place a mirror on the ! oor 4 feet from that wall. Each group member should take a turn standing on the spot 20 feet from wall and look into the mirror. Other group members should help the observer locate the point on the wall that the observer sees in the mirror, and then measure the height of this point above the ! oor. Before moving the mirror, each group member should take a turn as the observer. Repeat the same process by moving the mirror to locations that are 7 and 10 feet away from the wall as well. Use the table below to record your results.

Distance from wall to mirror

Height of the point on the wall re! ected in mirror

Similari-Teen Ally 1 Ally 2 Ally 3

4 feet

7 feet

10 feet

7. Measure the eye level height for each member of the group and record it in the table below.

Eye-Level Height of Each Hero

Similari-Teen Ally 1 Ally 2 Ally 3

8. Write the ratio of the eye-level height of Person A to the eye-level height of Person B and the ratio of the 4-feet data for Person A to the 4-feet data for Person B. Compare these two ratios. What appears to be true?


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes



SUGGESTED LEARNING STRATEGIES: Debriefing, Use Manipulatives, Discussion Group, Identify a Subtask

9. Would the same result occur if the ratio of the eye-level heights of Person C and Person D were compared to the ratio of their 4-feet data? Show your calculations.

10. If the eye-level height of a ! ve-year-old child observer was 3.5 feet, what data can you predict for the 4-feet row in the table in Item 6?

11. As the distance from the wall to the mirror increased in the table in Item 6, did the height above the " oor of the observed point on the wall increase or decrease? Explain why this occurs.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


SUGGESTED LEARNING STRATEGIES: Debriefing, Use Manipulatives, Discussion Group, Identify a Subtask

GMCongratulations, Similari-Teen. Your heat vision training is almost complete. Professor Phys-X will be helping you through the next phase of your lesson.

Professor Phys-XGreetings, Similari-Teen. I hear you’re about to try out your heat vision in the ! eld for the ! rst time. I’m sure you’ve already been warned of the potential dangers involved. Nonetheless, Dr. Protractor must be stopped, so we should get on with it.

" e wall of Dr. Protractor’s lab is a # at, level surface, and each ray of your heat vision that shines into the mirror re# ects at the same angle. " e scienti! c explanation is that the angle of incidence equals the angle of re# ection.

When you look into the mirror through the keyhole, the “line of sight” rays also behave in exactly the same way. In other words, the incoming line of sight forms an angle with the mirror, and the re# ected line of sight forms another angle. " e measure of each of these angles is equal.

Peer into the keyhole again, and estimate the distance between the door and the wall with the mirror.

Similari-TeenIt looks like it’s about twelve feet away.

Professor Phys-XVery well. Look again and tell me how far the device is from the wall with the mirror.

Similari-Teen" at’s a little more di$ cult to judge, but I’d say it’s about twenty-four feet from the wall to the device.

Professor Phys-XLook into the keyhole one last time, and tell me how far the mirror is from both the front wall and the back wall of the room.

Similari-Teen" e mirror is probably ! ve feet from the front wall of the lab and ten feet from the back wall.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes



SUGGESTED LEARNING STRATEGIES: Close Reading, Marking the Text, Activating Prior Knowledge, Discussion Group, Think/Pair/Share

12. Label the following diagram with the estimated measurements from Similari-Teen’s lesson with Professor-Phys-X. ! en use properties of similar triangles to determine the distances from the door to the mirror and from the mirror to the device.

Professor Phys-XA successful hit-roll is going to involve three rolls of the d20. If the sum of the two rolls is greater than or equal to the sum of the projected sum of the heat vision paths (the hypotenuses of the similar triangles), then you will be successful. Do you wish to attempt this now, or would you like to investigate some other possibilities ! rst?

Similari-TeenI would like to try now, but all this preparation has made me a little nervous.

Professor Phys-XIt’s OK to be nervous when using a new ability. In fact, it’s part of what makes you a hero.

CONNECT TO PROBABILITYPROBABILITY

The outcomes of many actions in tabletop RPGs are often decided by rolls of various-sided dice. The dice come in all shapes and sizes are referred to in the form d#. The 20-sided die is known as the d20.

Device

Back Wall

Front Wall Door

Mirror


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


SUGGESTED LEARNING STRATEGIES: Close Reading, Visualization, Discussion Group, Think/Pair/Share, Group Presentation, Use Manipulatives, Create Representations

To help Similari-Teen understand how the location of the mirror on the wall, at di! erent distances from the front to the back of the room, is related to the distance of the paths traveled by the heat vision rays, the professor sets up the following experiment. Similari-Teen stands 20 feet from the wall and places the mirror at various locations on the " oor along a line from his feet, which are perpendicular to the wall. Similari-Teen records his results from the experiment in a table like the one shown below in Item 13.

13. Show the results of Similari-Teen’s experiment. Use properties of similar triangles to calculate the measurements in the table below.

Distance from the Wall to the Mirror (in feet)

Height of the Point Above the Floor

(in feet)

3

7

11

15

14. Examine the data in the table in Item 13.

a. By what constant amount do the data in the # rst column increase?

b. Is there a constant increase in the values in the second column of the table? Explain your answer.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes



SUGGESTED LEARNING STRATEGIES: Discussion Group, Think/Pair/Share, Group Presentation, Create Representations

c. Graph the data from the table in Item 13. Explain why the relationship between the distance from the wall to the mirror and the height of the point above the ! oor is or is not linear. Be sure to label your graph appropriately.

y

x2018161412108642

2

4

6

8

10

12

14

16

18

20

15. If Similari-Teen stands 20 feet from the wall and the mirror is placed at an arbitrary distance of x feet from the wall, how far above the ! oor is the point that Similari-Teen sees re! ected in the mirror? Express the height of this re! ected point in terms of x. Explain how you arrived at this result and support your explanation with a drawing.


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes


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

16. a. How tall is the wall in your classroom? Where should the mirror be placed so that the re! ected light from the mirror will shine at the top of the wall?

b. What part of the wall would be seen if the mirror were placed directly atop Similari-Teen’s foot? Explain your reasoning.

GM! ere is nothing more you can do to prepare yourself, Similari-Teen. ! e time has come to destroy the device once and for all. Are you ready?

Similari-TeenI’m still a little nervous, but I know exactly what I need to do. Has Slide-Rule Girl determined the exact measurements?

GMIndeed she has. I’m transmitting her schematic to you now. It should show up on your graphing calculator display any second now. Good luck, Similari-Teen, we’re counting on you to save the day!

" e transmission from Slide-Rule Girl appears in the form of the diagram below.

22.5 ft.

8 ft.

12 ft.

15 ft.

Device

Back Wall

Front Wall Door

Mirror


© 2

010

Colle

ge B

oard

. All

righ

ts re

serv

ed.


My Notes



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

17. Help Similari-Teen destroy the device by completing the following:

a. Mark the diagram to indicate that corresponding angles are congruent.

b. Determine the scale factor of the similar triangles.

c. Use properties of similar triangles to determine the total distance that must be traveled by the rays of Similari-Teen’s heat vision.

Epilogue

GMCongratulations Similari-Teen. Dr. Protractor will certainly be coming home to a surprise tonight. I’m also pleased to tell you that your heat vision ability has increased by +2 … not to mention your similarity skills. ! ose have increased by a scale factor equal to that of the triangles created by your heat vision rays.

Professor Phys-XWe’re all very proud of you. I sense that your acts of heroism will be increasing exponentially as each day passes.

Similari-Teen! ank you both. I de" nitely couldn’t have done it without your help.

Professor Phys-XMathematics is a very powerful medium. It can lead you on some incredible journeys if you let it. Just remember, with great power comes great responsibility.

Similari-TeenHey, I think I’ve heard that somewhere before …

Professor Phys-XPerhaps. I’ve always thought it was such a marvel adage.


© 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. Determine if the following pairs of triangle are similar. If so, state the postulate or theorem that justi! es similarity, and write a similarity statement.

a.

A

B

E

D

C6!

6!

9!4!

b. YN

L

M

5 cm

7 cm

10.5 cmX

Z

14 cm

10 cm

21 cm

c. Use the meaning of similarity transformation to explain why the triangles in part b are similar or not similar.

2. Standing 8 feet from a puddle of water on the ground, Gretchen, whose eye height is 5 feet, 2 inches, can see the re" ection of the top of a " agpole. # e puddle is 20 feet from the " agpole. How tall is the " agpole?

3. Write a convincing argument to explain why !TUV ∼ !RSV.

T R V

S

U

4. Given: QR ___ VT = QS ___ VU = RS ___ TU

Prove: !QRS ∼ !VTU

T U

V

R S

Q

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

Compare and contrast the SAS similarity theorem for

triangles with the SAS congruence postulate for triangles.

A similarity transformation is a mapping in which the length of each side of a ! gure is multiplied by the same positive constant to produce a similar ! gure.

18. Is !JKL similar to !PQR? Explain your answer in terms of a similarity transformation.

J

L K12 16

20 2415 18

QR

P

205-216_SB_Geom_3-2_SE.indd 216205-216_SB_Geom_3-2_SE.indd 216 2/15/11 10:26:16 AM2/15/11 10:26:16 AM

similarity activity similari-teen saves the day 3-+unit+3... · similari-teen saves the day ... e...

Documents