Similarity, Right Triangles, 3 and Trigonometryclassroom.dickinsonisd.org/users/0315/docs/cb_sb_math_miu_l5_u3...Similarity, Right Triangles, Unit3 and Trigonometry Essential Questions

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191

Unit

3Similarity, Right Triangles, and Trigonometry

Essential Questions

How are similar triangles used in solving problems in every day life?

What mathematical tools do I have to solve right triangles?

Unit OverviewIn this unit you will study special right triangles and right triangle trigonometry. You will also study similar polygons, scale factors, and proportionality.

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

similar polygon sine scale factor

cosine

trigonometric ratio tangent Pythagorean theorem

This unit has three Embedded Assessments. These embedded assessments allow you to demonstrate your understanding of similarity, proportionality, special right triangles, and right angle trigonometry.

Embedded Assessment 1

Similarity in Polygons p. 229


The Pythagorean Theorem and Geometric Mean p. 243


Special Right Triangles and Trigonometry p. 267

EMBEDDED ASSESSMENTS

??

??

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192 Springboard® Mathematics with Meaning™ Geometry

Write your answers on notebook paper.

1. Simplify.a. √

___ 72

b. 7 ___ √__

5

2. Solve the following for x.

a. 5 = 2 __ x

b. x2 + 3x + 2 = 0

c. √__

2 ___ √__

7 = x ____ √

___ 14

3. Find the distance between (3, 5) and (7, –1).

4. Find the lengths AB and BD if AD = 29 units.

2x x2x + 4A B C D

5. Solve the following equation for a.

2a + b ______ c = d

6. Evaluate 4 √__

2 + 2( √___

36 - √__

8 ).

7. Find the length of side a in the right triangle pictured.

7

3a

8. In the fi gure below c is a transversal cutting parallel lines, l and m. List at least 4 geometric relationships that exist between angles 4, 5, and/or 8.

134

2

67 8

5

l

c

m

UNIT 3

Getting Ready

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Unit 3 • Similarity, Right Triangles, and Trigonometry 193

My Notes

ACTIVITY

3.1Exploring Similar FiguresPicture ThisSUGGESTED LEARNING STRATEGIES: Close Reading, Summarize/Paraphrase/Retell, Think Aloud

In magazines and newspapers, most advertisements and photographs are designed to fi t the width of the print column. Th e publishing industry uses the phrase column inch to describe a space that is one inch high and one column wide, regardless of the width of the column. A publication needs to be able to adjust images to fi t any column width. In the industry, there is a “trick of the trade” for rescaling advertisements and photographs, which uses the following approach.

Step 1 A diagonal is drawn from the lower left corner to the upper right corner of the image. No border or margin should be included in the image.

ColumnHeight

ColumnWidth

Step 2 Th e desired column width is measured from the lower left corner of the image, across the bottom of the image. Th is point is marked and a vertical line is drawn, perpendicular to the bottom edge, through this point. Th e perpendicular line is extended to the diagonal.

Step 3 A horizontal line is drawn from the point of intersection on the diagonal to the left edge of the image. Th e height of the new rectangle corresponds to the number of column inches of the image when it is resized to fi t the desired column width.

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194 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Exploring Similar Figures ACTIVITY 3.1continued Picture ThisPicture This

1. Follow steps 1, 2, and 3 in the Introduction on the photograph above, and complete the table below. You may fi nd it helpful to use a colored pencil when marking the photograph.

2. Label the lower left corner of the photograph A, the lower right corner F, and the points you marked on the bottom of the photograph B, C, D, and E (from left to right).

3. Label the points of intersection along the diagonal G, H, I, and J (from left to right) and label the upper right corner of the photograph K.

4. Measure the four acute angles formed by the diagonal and the vertical lines perpendicular to the bottom edge of the photograph. Record the measure of each angle in the table.

Column Width (in inches)

Column Height (in inches)

3.75

2.5

0.75

0.5

Column Width (in inches)


3.75

2.5

0.75

0.5

Angle Measure

∠AGB

∠AHC

∠AID

∠AJE

Angle Measure

∠AGB

∠AHC

∠AID

∠AJE

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


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My Notes

ACTIVITY 3.1continued

Exploring Similar Figures Picture ThisPicture This

5. What is the measure of each of the angles formed by the bottom of the photograph and the vertical lines drawn to the diagonal?

6. Explain how the angle measures of �ABG, �ACH, �ADI, �AEJ, and �AFK compare to one another.

7. Use the data collected in Item 1 to complete the table below. Be certain to express the ratios in simplest form.

Triangle 1

Triangle 2

Column Height of �1 ___________________ Column Height of �2

Column Width of �1 __________________ Column Width of �2

�ABG �ACH

�ABG �AEJ

�ACH �AFK

8. What patterns do you observe in the table in Item 7?

9. Choose a pair of triangles in the photograph in Part I that were not compared in Item 7.

a. Which two triangles did you choose?

b. Write the ratio of the heights of each triangle you chose, in simplest form.

c. Write the ratio of the widths of each triangle you chose, in simplest form.

SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Identify a Subtask, Think/Pair/Share, Look for a Pattern, Discussion Group


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My Notes


SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Identify a Subtask, Think/Pair/Share, Discussion Group, Quickwrite, Activating Prior Knowledge

10. Compare your results from Items 7–9 with other members of your group. Summarize your observations below.

11. What would you expect to discover if you chose any two of the triangles listed in Item 6 and calculated the ratio of the sides that lie along the photograph’s diagonal? Explain below.

12. Polygons are said to be similar if corresponding angles are congruent and corresponding sides are in proportion. A pair of similar triangles is denoted in the following form: �AFK∼�AEJ.

a. Refer to the triangles drawn on the photograph in Item 1. According to the defi nition of similarity, explain why �AFK is similar to �AEJ.

b. According to the defi nition of similarity, which of the triangles drawn on the photograph in Part I are similar? Explain below.

c. Th e scale factor of two similar polygons is the ratio formed by comparing a pair of corresponding sides. Determine the scale factor for �AFK and �AEJ.

ACADEMIC VOCABULARY

similar polygonspolygons with congruent corresponding angles and corresponding sides in proportion to one another

ACADEMIC VOCABULARY

scale factor


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My Notes



SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Interactive Word Wall, Notetaking,Discussion Group

13. Draw a smaller triangle similar to the one shown below. Explain how you created the triangle.

14. Determine the scale factor of the smaller triangle to the larger triangle in Item 13. Explain why it does not matter which pair of corresponding sides is selected in order to fi nd the scale factor.


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My Notes


SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Discussion Group, Predict and Confirm, Group Presentation

15. Recall the measurements found in Item1.

Column Width (in inches) 5 3.75 2.5 1.875 1.25Column Height (in inches) 2 1.5 1 0.75 0.5

a. Plot the ordered pairs (column width, column height) on the grid below.

5

4

3

2

1

1 2 3 4 5 6 7Column Width (in inches)

Colu

mn

Hei

ght (

in in

ches

)

b. Are these data linear? Explain below.

c. Use your graph to predict the height of the photograph examined in Item 1 if the width is increased to 6 inches. Explain how you arrived at your answer.


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My Notes



SUGGESTED LEARNING STRATEGIES: Identify a Subtask, Look for a Pattern, Discussion Group, Predict and Confirm, Summarize/Paraphrase/Retell

16. Complete the table below by writing the ratio of column height to column width in simplest form.

Triangle Column Width (in inches)


Ratio of Height to Width

�ABG 1.25 0.5

�ACH 1.875 0.75

�ADI 2.5 1

�AEJ 3.75 1.5

�AFK 5 2

a. Compare the ratios of height to width for each of the triangles and describe any patterns you observe.

b. Recall from Item 12 that all of the triangles formed by drawing the column width and column height are similar. Write a proportion that relates the column height y to the column width x for any triangle drawn on the photograph in this way.

c. Solve the proportion in part b for y.

d. Is this equation linear? Explain below.


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My Notes


SUGGESTED LEARNING STRATEGIES: Identify a Subtask, Look for a Pattern, Summarize/Paraphrase/Retell, Discussion Group, Quickwrite

17. Th e equation in Item 16 can be written as the function f (x) = kx, where x and f (x) are the column width and height, respectively. In the function, k is known as the constant of proportionality.

a. Rewrite the equation in Item 16b as a function.

b. What is the constant of proportionality for the function in Part a?

c. Graph the function in Part a on the coordinate axes shown in Item 15. How does the graph of f (x) compare to the data you previously plotted?

d. Explain the meaning of the constant of proportionality for the function in Part a in terms of column width and column height.

18. Use the function from Item 17a to calculate the column height of a photograph whose width is 2.5 inches. How does this answer compare to your answer in Item 1?

19. Use the function from Item 17a to calculate the column height of a photograph whose width is 6 inches. How does this answer compare to your answer in Item 15c?

CONNECT TO ARCHITECTUREARCHITECTURE

A well-known example of a constant of proportionality is the Golden Ratio. Mathematically, this

ratio is defi ned as 1 + √

__ 5 _______ 2 or

approximately 1.618. Figures constructed in this proportion are thought to be aesthetically pleasing and can be found in many works of architecture.


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My Notes



SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Discussion Group, Think/Pair/Share, Quickwrite

20. Based on the defi nition of similar fi gures in Item 12, would you consider the two images below to be similar? Explain why or why not.

D

BA

X YC

D′

B′A′

X′ Y′

C′

21. Measure AB and A'B' in centimeters and fi nd the ratio of AB to A'B' to the nearest tenth.

22. Measure CD and C 'D' in centimeters and fi nd the ratio of CD to C 'D' to the nearest tenth.

23. Without measuring them, make a conjecture about any two corresponding lengths, such as XY and X'Y'.

Th e Math Guy hosts a weekly television program on a local station. In one episode, he gives the following defi nition for similar fi gures: Figure A is similar to Figure B if, and only if, all of the distances between any two points on Figure A and the corresponding two points on Figure B are proportional.

24. Based on this defi nition, are the light bulbs similar? Explain.

READING MATH

In the fi gures to the left, points A and A' (pronounced “A prime”) correspond with one another. This notation is often used to assist in identifying corresponding parts of similar and congruent fi gures and the corresponding vertices of fi gures that have undergone a transformation such as a translation or refl ection.


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My Notes


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

25. Explain what is incorrect in the following argument, provided by an avid viewer of the television show. Th e viewer’s argument is based on the Math Guy’s defi nition of similar fi gures.

C

A

B D

4

6

4

6

B' D'

C'

A'

22

3 3

Since = = = = , the figures are similar.ABA'B'

BCB'C'

CDC'D'

DAD'A'

21

An Avid Viewer’s Interpretation of Similar Figures


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My Notes



SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Discussion Group, Think/Pair/Share, Predict and Confirm, Quickwrite

26. Consider the statement: All are similar. Fill in the blank with each term from the following list. Determine which statements are true and which statements are false (the fi gures are sometimes similar). Th en complete the table below by writing the fi gure that corresponds to the statement in the appropriate column.

Polygons Parallelograms Triangles Rectangles Right Triangles Squares Isosceles Triangles Circles Equilateral Triangles Rhombi Quadrilaterals Regular Hexagons

True False (sometimes similar)

27. Based on the table in Item 26, compare the properties of the fi gures that are always similar to those that are sometimes similar.


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My Notes


Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING


1. Th e following triangles are similar.

B

A CD

F

E

12 cm 10 cm7.2 cm 6 cm

9.6 cm16 cm

a. Determine the constant of proportionality of the triangles.

b. Write a similarity statement for the triangles.

2. Solve for x and y in the following fi gures.

52°

4x – 10° 38°

y°

3. Two similar triangles have a scale factor of 2 __ 3 . If the longest side of the smaller triangle is 15 inches, determine the length of the

longest side of the larger triangle.

4. Given: �FGH ∼ �JKL

a. List the pairs of corresponding sides of the triangles.

b. List the pairs of corresponding angles of the triangles.

5. Determine the values for x and y in the similar triangles shown below.

10 in.6 in.

4 in.

y°

21°

x 6. MATHEMATICAL

R E F L E C T I O N Amend the Math Guy’s defi nition of similar

fi gures in Item 24 so that it cannot be misconstrued by an avid viewer.

Another way to defi ne similarity is in terms of congruence. Th at is, two fi gures are similar if one of them is congruent to a rescaling of the other. For example, consider two squares, one with side lengths of 2 units and the other with side lengths of 4 units. If you rescale the square with side lengths of 2 units by multiplying the side length by 4 __ 2 ( 2 • 4 __ 2 = 4 ) , you get a square that is congruent to the square with side lengths of 4 units. Th erefore, the two squares are similar. Likewise, you can rescale the square with side lengths of 4 units ( multiply by 2 __ 4 ) to get a square that is congruent to the square with side lengths of 2 units.

28. Prove that all circles are similar by rescaling. Use the two general circles, one with a radius of a units and one with a radius of b units.

y

b

a

x

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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. Th eir game is a tabletop role playing game (RPG). A typical RPG involves players who create fi ctional characters to participate in imaginative stories. Th ese characters typically have specifi c abilities, which they use throughout the story to drive the action and outcome of the game. Th e use of these abilities is defi ned by a formal system of rules which is governed by a designated game master, or GM, for the session.

Th e game that Tristan and Shawnda created involves superheroes in an ongoing fi 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 fi rst prove the theorems involved in each scenario before applying their abilities to complete the tasks.

Th 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 refl ection of a strange device in the back left 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 refl ect the rays of my heat vision off the mirror and destroy the device.

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


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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 Th eorem.

Th e Side-Angle-Side (SAS) Similarity Th 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. Confi rm that the sides including ∠Q and ∠V are in proportion.

b. Draw point W on __

TV so that __

WV � __

RQ .

c. Th 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


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My Notes


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 Th eorem.

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

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

5. Help Similari-Teen complete his fi 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 fi 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 fi 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.


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My Notes


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

6. Locate a spot on the fl oor that is 20 feet away from one of the walls of the classroom. Place a mirror on the fl 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 fl 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 refl 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?


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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 fi 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 fl oor of the observed point on the wall increase or decrease? Explain why this occurs.


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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 fi eld for the fi 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.

Th e wall of Dr. Protractor’s lab is a fl at, level surface, and each ray of your heat vision that shines into the mirror refl ects at the same angle. Th e scientifi c explanation is that the angle of incidence equals the angle of refl 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 refl ected line of sight forms another angle. Th 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-TeenTh at’s a little more diffi 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-TeenTh e mirror is probably fi ve feet from the front wall of the lab and ten feet from the back wall.


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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. Th 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 fi 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


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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 diff 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 fl 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 fi rst column increase?

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


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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 fl 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 fl oor is the point that Similari-Teen sees refl ected in the mirror? Express the height of this refl ected point in terms of x. Explain how you arrived at this result and support your explanation with a drawing.


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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 refl 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.

GMTh ere is nothing more you can do to prepare yourself, Similari-Teen. Th 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!

Th 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


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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. Th 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-TeenTh ank you both. I defi 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.


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1. Determine if the following pairs of triangle are similar. If so, state the postulate or theorem that justifi 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 refl ection of the top of a fl agpole. Th e puddle is 20 feet from the fl agpole. How tall is the fl 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 fi gure is multiplied by the same positive constant to produce a similar fi 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


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ACTIVITY

3.3Triangle ProportionalityParallel Universe

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Close Reading, Marking the Text, Role Play, Visualization, Group Presentation, Think/Pair/Share, Create Representations, Identify a Subtask, Quickwrite, Self/Peer Revision

Parallel lines are used extensively in the world. Look around you. Chances are you will notice several examples. Window panes, wall supports, roof beams, lines in the road, streets in a city grid, picture frames, neon signs, stripes on fabric, and many more incorporate parallels. We have already studied several angle properties associated with parallel lines intersected by transversals. In this section we will explore segment lengths associated with parallel lines intersected by transversals.

1. In the fi gure, � � � MR || __

ST .Explain why �MAR ∼ �SAT.

2. Knowing that corresponding sides of similar triangles are proportional, complete this proportion:

AM ____ AS = AR ___ ?

3. If AM = 12 cm, MS = 9 cm, AR = 15 cm, determine RT. Show your work.

4. If MS = 8 cm, AR = 25 cm, RT = 10 cm, determine AM. Show your work.

S T

A

M R


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Triangle ProportionalityParallel UniverseParallel Universe

Triangle Proportionality Theorem

If a line parallel to a side of a triangle intersects the other two sides, then it divides them proportionally.

5. Complete the proof of the Triangle Proportionality Th eorem.

Given:

Prove: b __ a = c __ d

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Close Reading, Graphic Organizer, Marking the Text, Questioning the Text, Group Presentation, Think/Pair/Share, Create Representations, Identify a Subtask, Work Backward, Notetaking, Quickwrite, Self/Peer Revision

Statements Reasons

1. �PIE with || 1. Given

2. 2. If two parallel lines are cut by a transversal, the corresponding angles are congruent.

3. 3.

4. � ∼ �PIE 4.

5. b _____ b + a = c _____ c + d 5.

6. b(b + a)(c + d ) _____________ (b + a) = (b + a)(c + d )c _____________ (c + d) 6. Multiplication Property of Equality

7. b(c + d) = (b + a)c 7. Property of the Multiplicative Identity

8. 8. Distributive Property

9. bd = ac 9.

10. bd ___ ad = ac ___ ad 10.

11. 11.

P E

I

R

b c

a d

M


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Triangle Proportionality Parallel UniverseParallel Universe

6. State the converse of the Triangle Proportionality theorem.

7. Write a convincing argument about why the converse of the Triangle Proportionality Th eorem is true.

SUGGESTED LEARNING STRATEGIES: Close Reading, Questioning the Text, Think/Pair/Share, Quickwrite, Self/Peer Revision


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Triangle Proportionality ACTIVITY 3.3continued

SUGGESTED LEARNING STRATEGIES: Marking the Text, Interactive Word Wall, Summarize/Paraphrase/Retell, Role Play, Notetaking

Parallel Proportionality Theorem

If two or more lines parallel to a side of a triangle intersect the other two sides of the triangle, then they divide them proportionally.

TRY THESE

Given: __

RE || __

AT || __

IO || __

NS Determine each length. Show your work.

a. ET b. AI

c. AT d. OS

e. IO f. NS

Angle Bisector Proportionality Theorem

An angle bisector of a triangle divides the opposite side of the triangle into two segments that are in proportion to the adjacent sides.

CR A I N36 cm 24 cm

40 cm

12 cm

S

O

T

E

48 cm27 cm

CR A I N36 cm 24 cm

40 cm

12 cm

S

O

T

E

48 cm27 cm

Parallel UniverseParallel Universe

The Parallel Proportionality Theorem is a corollary of the Triangle Proportionality Theorem since it is proven directly from it.


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Triangle Proportionality Parallel UniverseParallel Universe

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Close Reading, Graphic Organizer, Marking the Text, Role Play, Visualization, Group Presentation, Think/Pair/Share, Create Representations, Identify a Subtask, Work Backward, Notetaking, Quickwrite, Self/Peer Revision

8. Complete the proof of the Angle Bisector Proportionality theorem

Given: �CAP, __

CM bisects ∠ACP

Prove: AM ____ MP = CA ___ CP

Statements Reasons 1. 1. Given

2. 2. Defi nition of angle bisector

3. Extend __

CA to locate point S so that CS = CP 3. Ruler Postulate

4. ∠S � ∠4 4.

5. m∠S + m∠4 + m∠3 =180° 5.

6. m∠4 + m∠4 + m∠3 =180° 6.

7. 2(m∠4) + m∠3 =180° 7.

8. m∠1 + m∠2 + m∠3 =180° 8. Angle Addition Postulate and Linear Pair Postulate

9. m∠2 + m∠2 + m∠3 =180° 9.

10. 2(m∠2) + m∠3 =180° 10.

11. 2(m∠2) + m∠3 = 2(m∠4)+ m∠3 11.

12. 12. Subtraction Property of Equality

13. 13. Division Property of Equality

14. __

CM ‖ __

SP 14.

15. AM ____ MP = CA ___ CS and AM ____ MP = CA ___ CP 15.

A

M

C

P

S

34

21


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Triangle Proportionality ACTIVITY 3.3continued



Given: � � � AT ‖ � � � EY ‖ � � � SB Complete each proportion with the appropriate measure.

B

E

A

UT

YS

1. EA ___ BE = TY ___ ? 2. AT ___ ? = UA ___ UB

3. AB ___ UA = ? ___ TU

Given the diagram with __

TH || __

IN || __

KE and segment measures as shown.

Determine the following measures. Show your work.

4. IK 5. IN

6. IT 7. TH

Given �RMP with angle bisector �� MA

8. Determine AR. Show your work.

9. Determine RP. Show your work.

10. Given the diagram with � � � BI ‖ � � � ST ‖ � � � EN , explain how to demonstrate that ES ___ SB = NT ___ TI


If you sketched thetriangle MPR above and

drew a line parallel to side RP that intersected side MR at point X, side MA at point Y and side MP at point Z, name two pairs of similar triangles formed.

H 4 cm 6 cm 8 cm

6 cm10 cm

TI

N ER

K

H 4 cm 6 cm 8 cm

6 cm10 cm

TI

N ER

K

R

24 cm

20 cm

36 cm

M

APR

24 cm

20 cm

36 cm

M

AP

O

B

S

E

R

V

A N

T

IO

B

S

E

R

V

A N

T

I

Parallel UniverseParallel Universe


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ACTIVITY

3.4Coordinate Proofs of Similarity and CongruenceSome Things Never ChangeSUGGESTED LEARNING STRATEGIES: Shared Reading

“Grandma Alice, tell me the story of Th e Similes again,” said Jodi.

“Well,” said her grandmother, “when I was growing up, my two best friends were Bertha Krump and Cindy Lou Mathers.”

“Why were you called Th e Similes?” asked Jodi even though she’d heard the story many times.

“We were inseparable,” explained Grandma Alice. “We dressed alike, talked alike, and even fi nished each other’s sentences.”

Jodi’s heart warmed at the familiar smile that crept across Grandma Alice’s face. “Like peas in a pod,” she said.

“Exactly,” beamed Grandma Alice. “Our math teacher used to tell us that all the time. Aft er a while, we got so used to being compared to one another, we started calling ourselves Th e Similes.”

“Didn’t you make up a math problem about the three of you once?” asked Jodi.

“Oh, I sure did,” said Grandma Alice. She leaned over in her chair, opened a small drawer in the end table, and pulled out a crumpled sheet of graph paper. Th e diagram had yellowed with age, and the edges had grown tattered, but the two triangles were still visible.

CONNECT TO LANGUAGE ARTSLANGUAGE ARTS

A simile is a fi gure of speech in which one thing is compared to another, generally using ‘like’ or ‘as.’

y

x

A′

C′

(9, 7)

A

C

B(-7, 2)

(-4, 3.5)

(-3, 1)B′

(3, 4)

(11, 2)

-2 2

2

4

4

6

6

8

8

10 12-4-6-8-10


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Coordinate Proofs of Similarity and Congruence ACTIVITY 3.4continued Some Things Never ChangeSome Things Never Change

“What is the problem?” asked Jodi.

“We were studying similar triangles in our Geometry class,” said Grandma Alice, “and I decided to plot our houses on a grid. Point A was my house on Prospect Street, Point B was Bertha’s house on Highland Avenue, and Point C was Cindy Lou’s house on Seville Road.”

“And what about the other triangle? Is it supposed to be similar to the fi rst one?”

Grandma Alice was a kind and generous woman, but she never let Jodi get through the story without doing the math herself. “You tell me, young lady,” she said.

1. In order to prove the triangles similar, Jodi can use the distance formula to test for SSS similarity. Calculate the distance between points to complete the chart below. Leave your answers in simplest radical form.

Segment Length

__

AB

__

BC

__

AC

__

A'B'

__

B'C '

__

A'C'

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Discussion Group, Think/Pair/Share


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Coordinate Proofs of Similarity and Congruence Some Things Never ChangeSome Things Never Change

2. Use your results from the table in Item 1 to compare the ratios of the corresponding sides of the two triangles.

AB ____ A'B' = BC ____ B'C'

= AC ____ A'C' =

3. Explain how your results from Item 2 prove the triangles are similar by the SSS Similarity Th eorem.

TRY THESE A

a. Determine if �DEF ∼ �TUV.�DEF = D(2, 1), E(4, 9), F (2, 13)�TUV = T(-6, 5), U(-2, 21), V(-6, 25)

b. Justify your answer to part a.

Grandma Alice beamed with pride. “I see you’ve been paying attention in Geometry class,” she declared.

“I sure have,” Jodi giggled. “In fact, we just learned another way to prove triangles similar.”

“Really? How exciting!” exclaimed Grandma Alice. “Show me!”

“I thought you’d never ask,” Jodi teased as she winked at her grandmother. “We can prove the same pair of triangles similar by the SAS Similarity Th eorem. Since we’ve already determined that the corresponding sides of the triangles are in proportion to one another, we can move ahead to proving a pair of corresponding angles congruent. To do this, we will explore the slopes of the corresponding sides.”

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


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

4. Complete the tables below by determining the slopes of the given sides.

�ABC �A'B'C 'Side Slope Side Slope

__

AB __

A'B'

__

AC __

A'C '

5. What do you notice about the slopes of corresponding sides? Justify your answer.

TRY THESE B

a. Use coordinate geometry to prove the following triangles similar.

W

U

EF

G

V

y

x4 6 8 102−2

−2

2

4

6

8

12

14

−4

−4−6−8

b. How did you decide which vertices in the triangles corresponded?

y

x

A′

C′

(9, 7)

A

C

B(-7, 2)

(-4, 3.5)

(-3, 1)B′

(3, 4)

(11, 2)

-2 2

2

4

4

6

6

8

8

10 12-4-6-8-10


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Coordinate Proofs of Similarity and Congruence Some Things Never ChangeSome Things Never Change

Th e same concepts can be expanded to other polygons as well. Consider rectangle JKLM below.

LM

8

10

6

4

2

–8–10 –6 –4 –2 2 4 6 8 10–2

–4

–6

–8

–10

y

x

J K

6. Plot QRST on the graph to create a rectangle similar to JKLM. Justify your answer by explaining the process you used to choose the coordi-nates of Q, R, S, and T.

Th e clanging of the grandfather clock in the corner startled them both. “Oh dear,” gasped Grandma Alice. “It’s fi ve o’clock already! You best be getting home before your mother sends out a search party for you.”

“What’s my assignment?” Jodi asked.

“I want you to explain why congruence is really just a special case of similarity,” she said as the tattered graph disappeared once again into the end table drawer. “Leave a note with your answer in my mailbox on your way home from school tomorrow, and we’ll talk about it the next time you come over to see me.”

“Will do,” said Jodi as she made her way across the living room toward the front door. “See you soon, Simile A,” she giggled.

SUGGESTED LEARNING STRATEGIES: Predict and Confirm, Discussion Group, Think/Pair/Share, Quickwrite


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TRY THESE C

a. Given points S and T, fi nd a third point, U, that would create a triangle similar to �PQR.

Q

P R

2

2

−2

−2

−4

−4 4

4

6 8 10 12 14

S T

b. Explain why Item a could have more than one solution.


1. Determine if �ABC ˜ �XYZ.

�ABC = A(4, 3), B(8, 3), C(4, 10)�XYZ = X(-1, 1), Y(-3, 1), Z(-1, 4.5)

2. Prove thetriangles similar.

3. Trapezoid QRST has vertices Q(4, 5), R(10, 5), S(9, 8) and T(5, 8). Trapezoid Q'R'S'T ' is being plotted so that it is similar to QRST.

a. Determine the coordinates for T ' if the other vertices are Q '(-13, 2), R'(-1, 2), and S '(-3, 8).

b. Determine the scale factor of corresponding sides of the two trapezoids.


Help Jodi with her “homework” by writing a

note to Grandma Alice that explains why congruence is a special case of similarity.

-22 4 6 8 10 12

18

16

14

12

10

8

6

4

2

-22 4 6 8 10 12

18

16

14

12

10

8

6

4

2



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Embedded Assessment 1 Use after Activity 3.4.Similarity in Polygons

A NEW RESOLUTION

Phaedra has fi nally saved up enough money from her aft er-school job to buy a new monitor. Th e fi rst time she boots up her computer aft er connecting the monitor, the icons on the screen are much larger than on her old monitor.

When she attempts to change her screen resolution, she’s presented with several choices: 640 × 480, 800 × 600, 1024 × 768, 1280 × 960, and 1600 × 1200.

Aft er some quick research on the internet, Phaedra thinks she’s discovered a solution. Several of the websites she visits mention somethingcalled Display Aspect Ratio (DAR), which is the same as a constant of proportionality. In most televisions and computer monitors, the standard DARis either 4:3 or 16:9. Phaedra knows that she has a 20 inch monitor, but shedoesn’t know which aspect ratio applies to her monitor. She decides to compare her resolution choices to determine this.

1. Convert Phaedra’s resolution choices to ratios in simplest form. Which constant of proportionality applies to her new monitor?

2. Phaedra’s new monitor is a 20 inch fl at screen model, which replaced her old 15 inch monitor. She knows that screen size is measured diagonally, so the measurements would be as follows:

12 in.

16 in.

20 in.

15 in.9 in.

12 in.

Write a convincing argument using similarity theorems/postulates that explains why the triangles are similar to one another.

CONNECT TO TECHNOLOGYTECHNOLOGY

In layman’s terms, a screen’s resolution can be interpreted as the number of pixels along the length and width of the display. A resolution of 1280 × 960 means that there are 1280 pixels rendered across the length of the screen and 960 pixels along the width.

New monitor icon

Old monitor icon

229-230_SB_Geom_3-EA-1_SE.indd 229229-230_SB_Geom_3-EA-1_SE.indd 229 1/21/10 2:19:42 PM1/21/10 2:19:42 PM

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Similarity in PolygonsA NEW RESOLUTION

3. Triangle NEW below represents half of Phaedra’s new monitor. Th e line y = 9 has been drawn to represent the height of her old monitor.

8

10

12

14

6

4

2

2 4 6 8 10 12 14 16 18 20

y

x

E

N W

S T

Write a convincing argument using similarity theorems/postulates that explains why �NEW ∼ �SET.

Embedded Assessment 1 Use after Activity 3.4.

Exemplary Profi cient Emerging

Math Knowledge# 1, 2, 3

The student:• Correctly converts the fi ve

resolution choices to ratios in simplest form and fi nds the correct constant of proportionality. (1)

• Selects the appropriate similarity theorems/postulates. (2, 3)

The student:• Correctly converts three or

four resolution choices to ratios in simplest form; may or may not fi nd the correct constant of proportionality.

• Selects some, but not all, of the appropriate theorems.

The student:• Correctly converts at least

one of the choices to simplest form; does not fi nd the correct constant of proportionality.

• Selects no appropriate theorems.

Communication# 2, 3

The student writes convincing arguments, using the appropriate similarity theorems/postulates, to explain why both pairs of triangles are similar. (2, 3)

The student writes a convincing argument for only one of the pairs of triangles.

The student writes a convincing argument for neither of the pairs of triangles.


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ACTIVITY

3.5Geometric MeanDo You Mean It?LEARNING STRATEGIES: Activating Prior Knowledge, Close Reading, Marking the Text, Questioning the Text, Think/Pair/Share, Create Representations, Notetaking, Self/Peer Revision

You have investigated properties and relationships of sides and angles in similar triangles. In this section you examine a special characteristic of right triangles.

Given the fi gure with right triangle MAE, ___ AN ⊥

___ ME , m∠M = 70°.

1. Determine these angle measures.

m∠MAN = m∠EAN = m∠E =

2. State why �MAN ˜ �AEN.

3. Th e large triangle is also similar to the two smaller triangles. Complete the similarity statement naming the large triangle appropriately.

�MAN ˜

4. Name the type of special segment AN is in relation to �MAE.

5. Given AN = 9 in. and NE = 12 in. Use the Pythagorean Th eorem and the properties of similar triangles to determine these segment lengths. Show your work.

AE = MA = MN =

Right Triangle Altitude Theorem

If an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original right triangle and to each other.

A

M N E


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Geometric Mean ACTIVITY 3.5continued Do You Mean It?Do You Mean It?

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Close Reading, Graphic Organizer, Marking the Text, Think/Pair/Share, Notetaking, Self/Peer Revision

6. Complete the proof of the Th eorem.

Given: �YEA with right angleEAY and altitude AS

Prove: �YEA ˜ �YAS ˜ �AES

Statements Reasons1. 1. Given

2. ∠ESA and ∠ASY are right angles 2.

3. ∠ESA � ∠ASY � ∠EAY 3.

4. ∠Y � ∠Y 4.

5. � ˜ � 5.

6. ∠E � ∠E 6.

7. � ˜ � 7.

8. �YEA ˜ �YAS ˜ � AES 8.

A

E S Y


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Geometric Mean Do You Mean It?Do You Mean It?

SUGGESTED LEARNING STRATEGIES: Close Reading, Interactive Word Wall, Marking the Text, Questioning the Text, Think/Pair/Share, Create Representations, Notetaking, Self/Peer Revision

7. Which two similar triangles allow us to say x __ t = t __ z ?

8. Which two similar trianglesallow us to say x __ u =

u __ p ?

Th e answers to Items 7 and 8 indicate two corollaries of the Right Triangle Altitude Th eorem.

Corollary 1: When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse.

9. In Item 7 we justifi ed that x __ t = t __ z .

Solve the proportion for t. Show your work.

TRY THESE

Given �AYE with altitude ___

AS (not to scale) as shown, solve the following. Show your work.

A

E S Y

ut

x z

w

p

a. x = 4 cm, z = 9 cm. Determine t.

b. x = 6 cm, t = 12 cm. Determine z.

c. z = 18 cm, t = 32 cm. Determine x.

A geometric mean of n numbers is the nth root of the product of the n factors.

Example:

30 is the geometric mean of 18, 20, and 75 since

30 = 3 √___________

18 · 20 · 75 .

MATH TERMS

A

E S Y

ut

x z

w

p


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Geometric Mean ACTIVITY 3.5continued Do You Mean It?Do You Mean It?

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

Corollary 2: When the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the part of the hypotenuse that is adjacent to the leg.

10. In Item 8 we justifi ed that x __ u = u __ p .

Solve the proportion for u. Show your work.

11. Write the proportion that can be written from corollary 2. Use w.

12. Solve the proportion for w. Show your work.



Write your answers on notebook paper. Show your work. 1. Determine the geometric mean of 21 and 84.

2. Determine the geometric mean of 16, 27, 54, and 72.

3. Th e fi gure at the right shows a side view of a garage. Th e roof hangs over the base portion by 2 feet in the back.Determine each of the following.

a. x b. z c. y

4. Given �KID as shown.

a. Determine ND.

b. Determine KD.

c. Determine KI.

5. Given the kite with the diagonal measures as shown.

a. Determine the length of the short diagonal.

b. Determine the side lengths.


Given �YEA with altitude

__ AS as shown; if

given any two of the variable measures, is it possible to determine all other measures? If it’s not always possible, state when it is and when it is not.

yx6

2

fron

t

z

K

I D26

24 N

24 in.6 in.

A

E S Y

ut

x z

w

p


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My Notes

ACTIVITY

3.6The Pythagorean Theorem and ItsConverse Is That Right?SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Marking the Text, Shared Reading, Summarize/Paraphrase/Retell

How did Pythagoras get a theorem named aft er him?Although many examples of the Pythagorean Th eorem were known

and used by the Babylonians, Chinese, Hindu and Egyptians well before Pythagoras was born (about 570 BCE), he is given credit for being the fi rst to formally prove the theorem. Many others since Pythagoras’ time, including a young man named James Garfi eld who would go on to be President of the United States, have also off ered formal proofs of the well known theorem.

Examine one proof of the Pythagorean Th eorem that is credited to Pythagoras himself.

Begin with a square having edges of length a + b. In the square, four right triangles with legs a and b have been drawn.

N OM

R

P

QS T

a b

b

a

ba

b a

1. Each of the four right triangles in the diagram above are congruent. What triangle congruence method justifi es this statement? Explain your answer.

2. Since the four right triangles are congruent, we know theirhypotenuses,

__ RN , __

TR , __

PT and __

NP , are congruent.

a. What reason can be used to justify this?

b. Label each hypotenuse in the diagram, c.

ACADEMIC VOCABULARY

THE PYTHAGOREAN THEOREMIn any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

If a and b are the lengths of the legs and c is the length of the hypotenuse then, c2 = a2 + b2.


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ACTIVITY 3.6continued Is That Right?Is That Right?

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Look for a Pattern, Quickwrite

3. �MNR is a right triangle and �MNR � �SRT.

a. What is the relationship between ∠MRN and ∠MNR?How do you know?

b. Use the congruence statement, ∠MNR � ∠SRT.What does this indicate about the relationship between ∠MRN and ∠SRT? Explain your reasoning.

c. What kind of angle is ∠NRT ? How do you know?

d. What are the measures of ∠RTP, ∠TPN and ∠PNR ? Justify your answer.

4. What special quadrilateral is formed by the four hypotenuses? Justify your answer.

The Pythagorean Theorem and Its Converse


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ACTIVITY 3.6continuedIs That Right?Is That Right?

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

5. It can be assumed from the diagram that the area of the large outside square is equal to the sum of the areas of the four triangles and quadri-lateral PNRT. Write an equation, in terms of a, b, and c that represents this statement.

6. Use algebraic properties to simplify both sides of the equation.

7. Solve the simplifi ed equation for c2.

You have now verifi ed algebraically, much as Pythagoras is thought to have done, Th e Pythagorean Th eorem and can use it to solve problems.

8. How high up a vertical wall will a 24 foot ladder reach if the foot of the ladder is placed 10 feet from the wall? Draw a sketch and show the calculations that support your answer.



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SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Close Reading, Marking the Text, Think/Pair/Share, Self/Peer Revision

9. Find the area of a rectangular rug if the width of the rug is 13 feetand the diagonal measures 20 feet. Draw a sketch and show the calculations that support your answer.

Th e Pythagorean Th eorem states that, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. Is the converse of this statement true?

10. Write the Pythagorean Th eorem in if-then form.

11. Write the converse of the Pythagorean Th eorem in if-then form.

12. Can the converse of the Pythagorean Th eorem be proven? Assume you have �ABC where c2 = a2 + b2, as shown below. Complete the following to try to prove �ABC is a right triangle. Use right �DEF, with legs a and b and hypotenuse f.

B

CA

a

b

c

E

FD

a

b

f



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SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Close Reading, Group Presentation, Quickwrite

a. It is known that f 2 = a2 + b2. Give a reason for this statement.

b. It was assumed that in �ABC, c2 = a2 + b2. So, the statementc = f can be made. Why is this true?

c. �ABC � �DEF by what reason?

d. ∠C is a right angle. Give a reason for this statement.

e. �ABC is a right triangle. What reason justifi es this statement?

13. You examined the converse of the Pythagorean Th eorem. Now, take a look at the inverse.

a. Write the inverse of the Pythagorean Th eorem in if-then form.

b. Is the inverse a true statement? Why or why not?



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SUGGESTED LEARNING STRATEGIES: Use Manipulatives

Since you have shown the Converse of the Pythagorean Th eorem is true, a little more exploration follows.

14. Use each of the following sets of triangle side lengths to build triangles using the manipulatives (straws) provided by your teacher.

Step 1: Cut manipulatives into 5 cm, 6 cm, 12 cm, 13 cm, and 15 cm lengths.

Step 2: Build each triangle on centimeter grid paper.

Step 3: Identify each triangle as right, acute or obtuse.

Step 4: Complete the table.

Triangle side lengths Type of triangle c 2 a2 + b2

5, 12, 13

6, 6, 12

5, 6, 12

5, 12, 15

5, 12, 12

6, 12, 13

6, 12, 15



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

15. What does your work in Item 14 suggest about the relationship between a2, b2, c 2 and the type of triangle?

16. Use the Converse of the Pythagorean Th eorem to determine whether each of the following sets of side lengths forms a right triangle. If a right triangle is not possible, tell whether an acute or obtuse triangle can be formed. Show the method you use to determine your answers.

a. 12, 34, 37

b. 6 __ 7 , 8 __ 7 , 10 ___ 7

c. 20, √___

42 , 21



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Write your answers on notebook paper or on grid paper. Show your work.

1. If a television screen is a rectangle with a 53 inch diagonal and a width of 45 inches, what is the height of the screen?

2. A standard baseball diamond is a square 90 feet on each side. Find the distance of a throw made from the catcher 3 feet behind home plate in an attempt to throw out a runner trying to steal second base. Round to the nearest whole number.

a. 93 feet b. 124 feet

c. 130 feet d. 183 feet

3. Tell whether a triangle can be formed having the following side lengths. If a triangle can be formed tell whether it is right, acute or obtuse.

a. 4, 6, 8 b. √__

8 , √__

8 , √___

16


Th e Pythagorean Th eorem was thought of by the early

Greeks as the following: Th e area of the square on the hypotenuse of a right triangle is equal to the sum of the areas of the squares on the legs.

Draw a diagram to illustrate this statement. Explain how your diagram illustrates the Pythagorean Th eorem.


SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge

Another way to prove the Pythagorean theorem is by using triangle similarity. In right triangle ABC below, an altitude is drawn to hypotenuse AB, forming two right triangles that are similar to triangle ABC.

Corresponding sides of similar triangles are in proportion, so you can write these proportions involving sides of the triangles.

b __ x = c __ b a ____ c – x = c __ a

17. Use the proportions above and algebra to prove a2 + b2 = c2.

A B

C

b

c

ah

x c – x


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The Pythagorean Theorem andGeometric MeanRIGHT TRIANGLE REGATTA

Cole is the fi nish line judge for the annual summer sailboat race on Lake Vacation. Before the race he helps the other judges set up the courses for the race. Each of the courses will be triangular in shape. Sailors will sail due west from the starting line, make a left turn around a buoy, creating a right angle, and sail due south to the fi nish line as in the diagram below.

Buoy

Finish

Start

Buoy

1. Th ere are three proposed triangular courses for the beginner sailors. Th e lengths of each leg of the race and the distance from the starting line to the fi nish line for each proposed course are listed below. Which course would be appropriate for the Right Triangle Regatta? Explain your answer, including reasons for not choosing the other two.

a. 20 miles, 26 miles, 34 miles

b. 16 miles, 30 miles, 34 miles

c. 15 miles, 31 miles, 34 miles

Th e course for the advanced sailors has already been decided. From the starting line, the boats will travel due west for 9 miles and then turn 90° around a marker buoy to travel due south 12 miles to the fi nish line. Th ere is a 2nd buoy located between the starting line and the fi nish line as shown in the diagram below.

Marker Buoy

Finish

Start

12 miles 2nd Buoy

9 miles

CONNECT TO NAVIGATIONNAVIGATION

A buoy is a fl oat anchored to mark a channel or a hazard under the surface of the water.


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The Pythagorean Theorem andGeometric MeanRIGHT TRIANGLE REGATTA

2. Cole must travel from the starting line to the fi nish line before the beginning of the race. What is the shortest distance from Start to Finish? Show the calculations that lead to your answer.

3. Th ere will be a judge positioned at each buoy and at the starting line and fi nish line.

a. How far must a judge travel from the starting line to the 2nd buoy? Show the calculations that lead to your answer.

b. Once the judge arrives at the 2nd buoy, how far will he be from the fi nish line? Show the calculations that lead to your answer.

4. What will be the distance between the two judges stationed on the two buoys? Show the calculations that lead to your answer.


Math Knowledge# 3a, b; 4

The student fi nds the correct three distances for the judges. (3a, b, 4)

The student fi nds only two of the correct distances.

The student fi nds fewer than two of the correct distances.

Problem Solving# 1, 2

The student:• Chooses the

most appropriate course. (1)

• Finds the correct shortest distance from Start to Finish. (2)

The student:• Uses a correct

method, but makes a computational error.

• Uses a correct method, but makes a computational error.

The student:• Exhibits a

misconception when trying to choose the course.

• Is not able to fi nd the shortest distance.

Communication# 1, 2, 3, 4

The student:• Gives a

complete, mathematically correct, explanation for his/her choice of the most appropriate course. (1)

• Shows correct calculations in determining the shortest course. (2)

• Shows correct calculations in fi nding the three distances. (3a, b, 4)

The student:• Gives an

incomplete explanation that contains no mathematical errors.

• Makes computational, not conceptual, errors.

• Makes computational, not conceptual, errors.

The student:• Gives an

erroneous explanation.

• Makes conceptual errors.

• Makes conceptual errors.


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ACTIVITYACTIVITY

My Notes

Special Right TrianglesThe Community Quilting ProjectSUGGESTED LEARNING STRATEGIES: Close Reading, Marking the Text, Summarize/Paraphrase/Retell, Activating Prior Knowledge

Th e Community Hospital wants to make its rooms more cheerful. Th e hospital asked volunteers to sew quilts for patient rooms and to decorate the common areas. Th e Hoover High Student Council wants to participate in the project. Ms. Jones, a geometry teacher, decides to have her classes investigate the mathematical patterns found in quilts.

Quilts are formed by sewing together three layers of fabric. Th e quilt top is the decorative top layer comprised of quilt blocks. Quilt blocks are oft en squares made up of smaller fabric pieces sewn together to create a pattern. Th e batting is a middle layer of padding. Th e bottom layer is typically a single piece of fabric called the backing. To fi nish a quilt, a quilter sews all three layers together using decorative stitches over and through the entire quilt area.

Th ere are many diff erent quilt block designs. Oft en these designs are named. Th e quilt block design made up of nine small squares, called the “Friendship Star” is shown.

1. Th e Friendship Star quilt block contains fi ve small squares and eight triangles.

a. Identify congruent fi gures in the quilt block and explain why they are congruent.

b. Classify the triangles in the quilt block by their angle measures.

c. Classify the triangles in the quilt block by their side lengths.

Some sample quilt block designs are shown below.

3.7

CONNECT TO AP

The special right triangle relationships are used to solve problems in trigonometry and calculus.


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The Community Quilting ProjectThe Community Quilting Project

SUGGESTED LEARNING STRATEGIES: Marking the Text, Summarize/Paraphrase/Retell, Create Representations, Use Manipulatives

d. What are the measures of the acute angles in each of the triangles? Explain your reasoning.

Part ITh e Hoover High Student Council decided to make Friendship Star quilts of various sizes with diff erent size quilt blocks. Small quilts will be made from 3-in. × 3-in. blocks, medium quilts from 4.5-in. × 4.5-in. blocks, and large quilts from 6-in. × 6-in. blocks. Ms. Jones’ classes will explore the dimensions of the triangles in each of these blocks.

2. Reproduce the Friendship Star quilt block design on the grid paper provided by your teacher.

a. Complete the appropriate row in the table below by measuring the dimensions of your entire quilt block and the lengths of the leg and hypotenuse of one triangle in the design you made.

b. Calculate the ratio of the hypotenuse to the leg of the triangle.Write the ratio as a decimal in the appropriate row in the table below.

Dimensions of Quilt Block

Length of Triangle Leg(in inches)

Length of Hypotenuse(in inches)

Ratio of Hypotenuse

to Leg

3 in. × 3 in.

4.5 in. × 4.5 in.

6 in. × 6 in.

c. Gather data from other students to complete the table.


Special Right Triangles


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Special Right Triangles The Community Quilting ProjectThe Community Quilting Project

SUGGESTED LEARNING STRATEGIES: Predict and Confirm, Look for a Pattern, Identify a Subtask, Quickwrite

3. What patterns do you notice in the table in Item 2?

4. Another Friendship Star quilt block is going to have 15-inch sides.

a. How can you determine the length of the legs of the triangleswithout measuring them?

b Using the patterns you observed in the table in Item 3, predict the length of the hypotenuse without measuring it.

c. What is the length of the hypotenuse of the right triangles in this size quilt block?

d. Is your answer to Part(c) the exact length of the hypotenuse?Why or why not?


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

5. Use the Pythagorean Th eorem to fi nd the exact values of the missing dimensions and ratios in the table below.

Dimensions of Quilt Block

Length of Triangle Leg(in inches)


Ratio of Hypotenuse

to Leg9 in. × 9 in.

15 in. × 15 in.1

1.52

6 √__

2 4


7. How close are the measured results in the table in Item 2 to the corresponding exact values in the table in Item 5?



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

8. Suppose that you are only given the length of one leg of an isosceles right triangle (45°– 45°– 90°).

a. Write a verbal rule for fi nding the lengths of the other two sides.

b. Let l be the length of the leg of any isosceles right triangle (45°– 45°– 90°). Use the Pythagorean Th eorem to derive an algebraic rule for fi nding the length of the hypotenuse, h, in terms of l.

9. Ms. Jones designed a variation on the Friendship Star quilt block, called the “Twisted Star.” Using the rules you derived in Item 8, what are the dimensions of the smallest triangle on the 12-in. × 12-in. quilt block shown below? Explain how you found your answer.

Special Right TrianglesThe Community Quilting ProjectThe Community Quilting Project

CONNECT TO AP

The special right triangle relationships are used to solve problems in trigonometry and calculus.

The length of the sides of each of the 9 small squares is 12 ___ 3 = 4 inches.

So the legs of the larger triangles are 4 inches long. Each leg of the smallest triangle is half the length of the leg of the larger triangle or 2 inches. Using the rule from item 8, l √

__

2 = h, so 2 √__

2 = h and the length of the hypotenuse is 2 √

__

2 inches.

Answers may vary. Sample rule: The other leg is equal in length to the given leg and the length of the hypotenuse is the given leg multiplied by √

__

2 .

l 2 + l 2 = h 2

2l 2 = h 2

√

___

2l 2 = √___

h 2

l √__

2 = h


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Special Right Triangles ACTIVITY 3.7continued The Community Quilting ProjectThe Community Quilting Project

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Marking the Text, Create Representations, Quickwrite

Ms. Jones introduces her class to a quilt block in the shape of a hexagon. Th is block is formed from six equilateral triangles, divided in half along their altitudes. Ms. Jones’ students know that if an altitude is drawn in an equilateral triangle, two congruent triangles are formed. Th e resulting hexagonal quilt block is shown below.

10. What is the measure of each of the angles in any equilateral triangle?

11. What are the measures of each of the angles in the smallest triangles in the hexagonal quilt block shown above? Explain your answer.


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12. Th e smallest triangles are special scalene right triangles. Th ey are oft en called 30°– 60°– 90° right triangles.

a. How can you determine which leg is shorter and which leg is longer using the angles of the triangle?

b. If each of the sides of the hexagonal quilt block is 4 inches, how long is the shorter leg in the 30°– 60°– 90° right triangle? Explain your answer.

c. What is the relationship between the length of the hypotenuse and the length of the shorter leg in a 30°– 60°– 90° triangle? Explain your answer.

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


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Special Right Triangles ACTIVITY 3.7continued The Community Quilting ProjectThe Community Quilting Project

SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern

13. Look for patterns between the longer leg of the 30°– 60°– 90° right triangle and the other sides by completing the table below using the Pythagorean Th eorem. Write each ratio in simplest form. Leave all answers in radical form.


Length of Shorter Leg(in inches)

Length of Longer Leg(in inches)

Ratio of Length of Longer Leg to Length

of Shorter Leg684

3.51


15. Suppose that you are only given the length of the shorter leg of a30°– 60°– 90° right triangle.

a. Write a verbal rule for fi nding the length of the longer leg.

b. Write a verbal rule for fi nding the length of the hypotenuse.


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SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Quickwrite

c. Let s be the length of the shorter leg of any 30°– 60°– 90° right triangle. Use the Pythagorean Th eorem to derive an algebraic rule for fi nding the length of the longer leg, l, in terms of s.

16. Use your work from Items 14 and 15 to determine the height of the hexagon block shown below, whose sides are 4 inches, if the height is measured from the midpoint of one side to the midpoint of the opposite side. Explain below how you found your answer.


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1. In a 45°– 45°– 90° triangle, if the length of a leg is 6 cm, the length of the hypotenuse is:

a. 12 cm b. 6 √__

3 cm

c. 6 √__

5 cm d. 6 √__

2 cm

2. Find a and b in the diagram below. Th e triangle is not drawn to scale.

45˚

b

a

7√2 cm

3. Square MNOP has a diagonal of 12 inches. Find the length of each side of the square.

4. Find a and b.

30˚

42

a

b

5. Find a and c.

10

a

c60˚

6. Find m.

8

60˚

m

a. 4 b. 4 √__

2

c. 4 √__

3 d. 8 √__

3

7. A ladder leaning against a house makes an angle of 60° with the ground. Th e foot of the ladder is 7 feet from the house. How long is the ladder?


Brayden’s teacher asked him to draw a 30°–60°–90°

right triangle. He drew the diagram shown. Tell why it is not possible for Brayden’s triangle to exist.

512

13

60˚30˚



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ACTIVITY

3.8Basic Trigonometric RelationshipsThe Sine of Things to ComeSUGGESTED LEARNING STRATEGIES: Graphic Organizer, Activating Prior Knowledge, Marking the Text

Tricia is a commercial artist working for Th e Right Angle Company. Th e company specializes in small business public relations. Tricia creates appealing logos for client companies. In fact, she helped create the logo for her company. Th e Right Angle Company will use its logo in diff erent sizes for stationery letterhead, business cards, and magazine advertisements. Th e advertisement and stationery letterhead-size logos are shown below.

THE

RIGHT

THE

RIGHT

Magazine Advertising-Size Logo Stationery Letterhead-Size Logo

1. Measure the two acute angles and the lengths of the sides of each logo above. Measure the angles to the nearest degree and the sides to the nearest tenth of a centimeter. Be as accurate as possible. Record the results in the table below.

Logo Size Hypotenuse LongerLeg

ShorterLeg

Larger AcuteAngle

Smaller AcuteAngle

Advertisement

Letterhead


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Basic Trigonometric RelationshipsACTIVITY 3.8continued The Sine of Things to ComeThe Sine of Things to Come

2. Although measurements can never be exact, the lengths of the sides of any right triangle satisfy the Pythagorean Th eorem. Confi rm that the Pythagorean Th eorem is satisfi ed by the measurements of the two right triangular logos on the preceding page. Show your work and results, but allow for some error due to measurement limitations.

3. Th e logos are similar triangles. Justify this statement. Th en give the scale factor of advertising logo lengths to corresponding letterhead logo lengths.

4. Th e triangular logo used on Th e Right Angle Company business cards is also similar to the logos used for advertisements and letterheads. Th e scale factor of letterhead logo lengths to corresponding business card logo lengths is 2.3:1.

a. Determine the length of each side of the business card logo.

b. Use a ruler to draw the business card logo to scale in the space below.

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations


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Basic Trigonometric Relationships The Sine of Things to ComeThe Sine of Things to Come

Tricia tries to incorporate a right triangle into many of the logos she designs for her clients. As she does, Tricia becomes aware of a relationship that exists between the measures of the acute angles and the ratios of the lengths of the sides of the right triangles.

5. Use each grid below to draw a right triangle that has a longer vertical leg of L units and a shorter horizontal leg of S units. Th e fi rst triangle is drawn for you. Use the Pythagorean Th eorem to fi nd the length H of the resulting hypotenuse to the nearest tenth. Record its length in the appropriate place at the bottom of each grid.

Grid A Grid B Grid CL = 8, S = 6 L = 5, S = 4 L = 4, S = 3

H = H = H =

Grid D Grid E Grid FL = 10, S = 8 L = 12, S = 9 L = 6, S = 3

H = H = H =

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Graphic Organizer, Marking the Text



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6. Some of the triangles in Grids A–F are similar to each other. Identify the groups of similar triangles, using the grid letter, and explain below how you know they are similar. You should fi nd a group of three similar triangles and a group of two similar triangles.

In any right triangle, the hypotenuse is opposite the right angle. For each acute angle, one of the right triangle’s legs is known as that angle’s opposite leg and the remaining leg is known as that angle’s adjacent leg. In �CAR below, the hypotenuse is

___ AC . For acute ∠C, side

___ AR is its

opposite leg and side ___

RC is its adjacent leg. For acute ∠A, side ___

RC is its opposite leg and side

___ AR is its adjacent leg.

C

R A

7. In right �BUS, identify the opposite leg and the adjacent leg for ∠U.

U

S B

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Close Reading, Interactive Word Wall, Think/Pair/Share


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8. One group of similar triangles, identifi ed in Item 6, is shown on the grids below. For each right triangle, the angle opposite the longer leg has been named with the same letter as the grid. Determine the ratios in the table and write the ratios in lowest terms.

SUGGESTED LEARNING STRATEGIES: Create Representations

Grid A Grid C Grid EL = 8, S = 6 L = 4, S = 3 L = 12, S = 9

A C E

H = 10 H = 5 H = 15

Length of Opposite Leg

___________________ Length of Hypotenuse Length of Adjacent Leg

___________________ Length of Hypotenuse Length of Opposite Leg

___________________ Length of Adjacent Leg

∠A

∠C

∠E


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My Notes


9. For each of the triangles in Item 8, use your protractor to fi nd the measure of the larger acute angle.

∠A ∠C ∠EMeasure of Larger Acute Angle

10. In Item 9, you found that the measures of each of the three angles were the same. If, in another right triangle, the measure of the larger acute angle was the same as the measures of ∠A, ∠C, and ∠E, what would you expect the following ratios to be?

a. length of opposite leg

__________________ length of hypotenuse =

b. length of adjacent leg


c. length of opposite leg

__________________ length of adjacent leg =

11. Explain how you reached your conclusions in Item 10.

SUGGESTED LEARNING STRATEGIES: Graphic Organizer, Look for a Pattern, Quickwrite


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Th e ratio of the lengths of two sides of a right triangle is a trigonometric ratio. Th e three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated sin, cos, and tan.

12. Use a scientifi c or graphing calculator to evaluate each of the following to the nearest tenth. Make sure your calculator is in DEGREE mode.

a. sin 53° =

b. cos 53° =

c. tan 53° =

13. In each column of the table in item 8, the ratios that you wrote were equal. Express the ratios from the three columns as decimal numbers rounded to the nearest tenth.







14. Compare your answers to Items 12 and 13. Th en describe each of the ratios below in terms of sin, cos, and tan. Assume that the ratios represent sides of a right triangle in relation to acute ∠X.







SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Think/Pair/Share

ACADEMIC VOCABULARY

trigonometric ratio

ACADEMIC VOCABULARY

sinecosinetangent

CONNECT TO AP

Trigonometric ratios are used to defi ne functions that model periodic phenomena like the hours of daylight.


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SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Close Reading, Interactive Word Wall, Marking the Text, Create Representations

Now that we have defi ned the sine, cosine, and tangent of an acute angle in terms of the sides of a right triangle we can defi ne three other trigonometric ratios. Th ese are the reciprocals of sine, cosine, and tangent. Cosecant (csc) is the reciprocal of the sine ratio, secant (sec) is the reciprocal of cosine, and cotangent (cot) is the reciprocal of tangent.

15. Use your knowledge of the sine, cosine, and tangent ratios to write the following ratios in terms of the sides of a right triangle. Assume that the ratios represent sides of a right triangle in relation to acute ∠X.

a. csc X =

b. sec X =

c. cot X =

TRY THESE

a. For �ABC in the My Notes section, write the ratios in lowest terms.

sin A = csc A = sin C = csc C =

cos A = sec A = cos C = sec C =

tan A = cot A = tan C = cot C =

b. Compare and contrast the ratios for ∠A and ∠C.

A

B C

15 17

8


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For each of the following triangles determine the ratios requested. Th en use a scientifi c or graphing calculator to evaluate each trigonometric function to the nearest thousandth and solve each equation for y. Round fi nal answers to the nearest tenth.

c. P

R y Q

41˚

49˚

100

d. 62˚

M N

O

y 18

28˚

sin 41° = cos 28° =

16. Use your knowledge of trigonometric functions to fi nd the value of x in �ABC in the My Notes section.

a. Choose an acute angle in �ABC.

b. Identify sides as opposite, adjacent, or hypotenuse with respect to the acute angle chosen.

c. Use the sides to choose an appropriate trigonometric function.

d. Write an equation using the identifi ed sides, acute angle, and trigonometric function chosen.

e. Solve for x.

SUGGESTED LEARNING STRATEGIES: Quickwrite

A

BC

x15

20˚


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SUGGESTED LEARNING STRATEGIES: Create Representations, Identify a Subtask

17. Use your knowledge of trigonometric functions to fi nd the value of y in the triangle below.

D

E

Fy

12

10˚

a. Choose an acute angle in �DEF and identify sides as adjacent, opposite, or hypotenuse with respect to the angle you chose.

b. Use the sides to choose an appropriate trigonometric ratio and write an equation using the identifi ed sides, acute angle, and trigonometric function.

c. Solve for y.


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SUGGESTED LEARNING STRATEGIES: Create Representations, Identify a Subtask

18. Tricia did such an exceptional job creating logos that she was given the task of making a banner and representing her company at a job fair. When Tricia got to the job fair, she was relieved to see there was a ladder she could use to hang the banner. While Tricia waited for someone to help her, she leaned the 12-foot ladder against the wall behind the booth. Th e ladder made an angle of 48° with the fl oor.

a. Use the information above to draw and label a right triangle to illustrate the relationship between the ladder and the wall.

b. Set up and solve an equation to fi nd how far up the wall the top of the ladder reaches.


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1. a. Find the missing measures in the given triangle.

A

B C

5

1222.6˚

b. Draw and label a triangle similar to the triangle given in part (a). Include each side length and angle measure.

c. State the scale factor of the triangle given in part (a) to the triangle you drew in part (b).

2. In �PQR, identify the hypotenuse, adjacent leg, and opposite leg for ∠R.

Q

R

P

3. Use your calculator to evaluate the following. Round to 3 decimal places.

a. cos 54° b. sin 12° c. tan 67°

4. Find each of the following ratios. Write each ratio in lowest terms.

25

N

O

M7

a. sin M b. cos O

c. tan O d. sec M

e. csc O f. cot M

5. Find x and y. Round fi nal answers to tenths.

y

x

18

42˚


Compare the values of the sine and cosine ratios as the

measure of an angle increases from 0° to 90°.


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Special Right Triangles and TrigonometryINCLINED TO BE SAFE

Th e Student Council at P. T. High School is participating in a building team for its yearly service project. Students who sign up must attend a series of training sessions.

Th e fi rst session focuses on ladder safety. Selma House urges the students to be aware of the angle a ladder forms with the ground when it is leaned against a wall or a roof. A ladder that is too steep or too fl at can be hazardous.

Selma tells the students that the range of safe angle measures for ladders to form with the ground is 50° to 75°.

1. Th ere are 18 foot ladders available at the building site. What is the minimum distance the foot of the ladder should be placed from the base of a wall? Explain your thinking. Include a diagram and calculations to support your explanation.

2. What is the maximum distance the foot of the ladder should be placed from the base of a wall? Explain your thinking. Include a diagram and calculations to support your explanation.

3. What is the maximum height on a wall that the ladder should reach in order to maintain safety? Explain your thinking. Include a diagram and calculations to support your explanation.

Emerson needs to access the roof of a storage shed at Selma House’s home. She has a 15 foot ladder and no calculator. As a result, she chooses to place her ladder safely using what she knows about 30°– 60°– 90° special right triangles.

4. To guarantee safety, exactly how far should she place the foot of the ladder from the base of the shed?

5. What is the height of the top of the ladder when it is placed to guarantee safety? Give an exact answer.


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Special Right Triangles and TrigonometryINCLINED TO BE SAFE


Problem Solving#1, 2, 3, 4, 5

The student:• Finds the correct

minimum and maximum distance the foot of the ladder should be placed from the wall. (1, 2)

• Finds the maximum height that the ladder should reach. (3)

• Finds the correct distance of the ladder from the base of the shed. (4)

• Finds the correct height of the top of the ladder. (5)

The student:• Finds either the

correct minimum or correct maximum distance, but not both.

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

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

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

The student:• Finds neither the

correct minimum nor correct maximum distance.

• Uses an incorrect method to fi nd the height.

• Uses an incorrect method to fi nd the distance.

• Uses an incorrect method to fi nd the height.

Representations#1, 2, 3

The student makes diagrams that correctly represent each of the three situations. (1, 2, 3)

The student makes only two correct diagrams.

The student makes fewer than two correct diagrams.

Communication#1, 2, 3

The student:• Gives a complete,

mathematically correct, explanation for the minimum and maximum distances. (1, 2)

• Gives a complete, mathematically correct, explanation for the maximum height. (3)


correct explanation for only one of the distances.

• Gives an incomplete explanation that contains no mathematical errors.


correct explanation for neither of the distances.

• Gives an incorrect explanation.


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UNIT 3Practice


ACTIVITY 3.1

1. Explain why the following triangles are similar to one another.

yy

x z x z

3n4n

7n

34

7

2. Th e following triangles are similar. Determine the values of x, y, and z.

X

15 + x

z - 18°

W YQ S

R

20 m

32 m

30 m

24 m

y

40°

3. Sketch a fi gure that is similar but not congruent to a rectangle with length 14 in. and width 8 in.

ACTIVITY 3.2

4. Solve for x in the following fi gure.

22 ft.

18 ft.

4x + 1

7.5 ft.

5. State the theorem or postulate that allows you to determine similarity in the following pairs of triangles.

a.

24 cm 12 cm

36 cm

12 cm

4 cm8 cm

b.

14 m5 m

c. The arrows indicate

parallel lines.

ACTIVITY 3.3

6. Given the diagram with ___

LD ‖ ___

AE ‖ ___

NT and segment measures as shown, determine the following measures. Show your work.

S D E

45 cm

48 cm60 cm

8 cm

16 cmA

L

T

N

a. SL b. LD c. ET d. NT

7. Given �CEL with measures as shown, determine x. Show your work.

UC

L

E36 cm48 cm

42 cmx

ACTIVITY 3.4

8. Th e ratio of similarity of corresponding sides of �STU and �VWX is 2 __ 3 . If the coordinates of S, T, and V are S (-4, 6), T(2, 6), and V(5, 7), determine a possible pair of coordinates for W.

9. Determine if �HIJ ˜ �LMN. �HIJ = H(8, -2), I(12, -2), J(-9, 8) �LMN = L(-8, -8), M(-6, -8), N(-8, -1)

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UNIT 3 Practice


ACTIVITY 3.5

10. Given �RAM as shown.

a. Determine PM. b. Determine RP. c. Determine RA.

P

R

M

A

8 in 10

ACTIVITY 3.6

11. Find the length of the hypotenuse of an isosceles right triangle with leg length 5 centimeters. Give the exact answer.

12. Find the length of the altitude drawn from the vertex of an isosceles triangle with side lengths 13 in., 13 in., and 24 in.

13. An isosceles trapezoid has bases that are 7 inches and 13 inches long. Th e height of the trapezoid is 4 inches. Find the perimeter of the trapezoid.

14. Tell whether a triangle can be formed having the following side lengths. If a triangle can be formed, tell whether it is right, acute, or obtuse.

a. 9, 40, 41 b. 4, 5, 6 c. 5, 12, 18 d. 9, 9, 13 e. 27, 36, 45

15. Use the given vertices to determine whether �ABC is a right triangle. Explain your reasoning and show the calculations that led to your answer. A(2, 7) B(3, 6) C(–4, –1)

ACTIVITY 3.7

16. Find a and b.

ab

45°

3

17. Th e measure of each leg of an isosceles right triangle is 5. Find the measure of the hypotenuse.

18. Th e perimeter of a square is 40 cm. Find the length of a diagonal.

19. Find the perimeter of a square, to the nearest tenth, if the length of its diagonal is 14 inches.

20. Find d and e.

e

d

860°

21. Th e longer leg of a 30°– 60°– 90° triangle is 6 inches. What is the length of the hypotenuse?

22. Th e length of an altitude of an equilateral triangle is 2 √

__

3 inches. Find the length of a side of the triangle.

23. One side of an equilateral triangle is 8 cm. Find the length of the altitude.

24. Th e perimeter of an equilateral triangle is 36 inches. Find the length of an altitude.

25. Find the perimeter of the trapezoid.

60°

18 cm

6 cm

26. Find the perimeter of �PQR.

R

P Q

60°

30°10 √ 3


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UNIT 3Practice


27. Find a and b.

6 b

a30°45°

ACTIVITY 3.8

28. Draw �RST with right angle S. Identify the hypotenuse, adjacent leg, and opposite leg for ∠T.

29. Use your calculator to evaluate the following.Round to 3 decimal places.

a. cos 49° b. sin 75° c. tan 17°

30. Using �ABC below, write a ratio for the following.

a. cos B. b. tan A c. csc B 17

15C B

A

31. In the diagram below, which trigonometric ratio corresponds to 4 __ 5 ?

D

E

F

5

4

3

a. cos E b. sin D c. tan D d. cos D

32. Label the 30°– 60°– 90° triangle shown below with hypotenuse length 8 cm and label side lengths. Use your triangle to fi nd the exact value of each ratio. Simplify all radicals.

a. cos 30°

30°

b. sin 30°

c. tan 30°

d. sin 60°

e. cos 60°

f. tan 60°

33. In �RST with right angle R, if sin T = 4 __

5 ; fi nd cos T and tan T.

34. Find x and y. Round to the nearest tenth.

36°

12y

xM N

O

35. Find x and y. Round to the nearest tenth.

60°

10

R P

y

x

Q

36. Find each of the following using the given triangle. Round to the nearest tenth.

a. m∠C 70°

7.2BC

A

b. AB c. AC

37. A kite string is 200 meters long. Find the height of the kite if the string makes an angle of 38° with the ground.

38. A ramp at the loading dock of an automobile manufacturing plant makes a 29° angle with the ground. Th e bottom end of the ramp is 30 meters from the building.

a. Draw and label a diagram to illustrate the situation.

b. Set up and solve an equation to find the length of the ramp. Round to tenths.

39. In an isosceles triangle, the base is 32 cm long and the base angles are 56°. How long are the legs?


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UNIT 3 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.

How are similar triangles used in solving problems in everyday life?

What mathematical tools do I have to solve right triangles?

Academic Vocabulary

2. Look at the following academic vocabulary words:

similar polygon scale factor trigonometric ratiosine cosine tangentPythagorean Th eorem

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?

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Unit 3

Math Standards Review

1. Marci wants to build a frame for her artwork in the shape of a triangle. She has three pieces of wood of lengths 4 meters, 7 meters, and 10 meters to use for the frame. Marci will use the entire length of each of the three pieces to make the frame. What type of triangle will she create?

A. acute triangle

B. equilateral triangle

C. right triangles

D. obtuse triangle

2. José is designing a fl oor plan. José wants his bedroom to be 12 feet by 15 feet. On the fl oor plan, the 12-foot length is represented by a segment which is 3 cm long. What is the length of the segment that represents the 15-foot length?

3. Rosa has a 12-foot ladder that she has put against the side of her house. Th e ladder rests against the bottom of a window that is 8 feet off the ground. Determine the measure of the angle the ladder makes with the ground to the nearest degree.

1. Ⓐ Ⓑ Ⓒ Ⓓ

2.

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

3.

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

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Unit 3 (continued)

Math Standards Review

4. Rodney is framing his high school diploma. He wants the frame tight around his diploma which is a square that is 10 inches on each side. Each wood frame piece is 2 inches wide and is shaped like this:

Part A: Determine the outside perimeter of the fi nished frame.

Part B: To make the frame stronger, Rodney will brace the frame with one diagonal piece of wood. Th e brace will fi t inside the square frame. Find the length of the longest part of the cross-brace which is shaped like this:

Part C: Explain in your own words how you solved Parts Aand B.

Read

Explain

Solve

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Similarity, Right Triangles, 3 and Trigonometryclassroom.dickinsonisd.org/users/0315/docs/cb_sb_math_miu_l5_u3...Similarity, Right Triangles, Unit3 and Trigonometry Essential Questions